%------------------------------------------------------------------------------
% File     : ITP281_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Functional 00357_014232
% 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    : 0077_VEBT_Intf_Functional_00357_014232 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11804 (4204 unt;1697 typ;   0 def)
%            Number of atoms       : 17998 (8416 equ)
%            Maximal formula atoms :   39 (   1 avg)
%            Number of connectives : 19964 (2029   ~; 317   |;2042   &)
%                                         (2073 <=>;13503  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   31 (   2 avg)
%            Number of FOOLs       : 1368 ( 637 fml; 731 var)
%            Number of X terms     : 1067 (   0  []; 801 ite; 266 let)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  : 1426 (1271   >; 155   *;   0   +;   0  <<)
%            Number of predicates  :  210 ( 207 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1504 (1504 usr;  89 con; 0-7 aty)
%            Number of variables   : 34651 (31130   !; 819   ?;34651   :)
%                                         (2702  !>;   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
%            2022-02-18 17:39:40.323
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (1680)
tff(sy_cl_HOL_Otype,type,
    type: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_GCD_Oring__gcd,type,
    ring_gcd: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Complete__Lattices_OSup,type,
    complete_Sup: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
    semiri6843258321239162965malize: 
      !>[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_Euclidean__Division_Oeuclidean__ring,type,
    euclid5891614535332579305n_ring: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

tff(sy_cl_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_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_Limits_Otopological__comm__monoid__mult,type,
    topolo4987421752381908075d_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
    real_V5047593784448816457lgebra: 
      !>[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__boolean__algebra,type,
    comple489889107523837845lgebra: 
      !>[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__ring__with__nat,type,
    euclid8789492081693882211th_nat: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__alr____,type,
    aTP_Lamp_alr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__als____,type,
    aTP_Lamp_als: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alt____,type,
    aTP_Lamp_alt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alu____,type,
    aTP_Lamp_alu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alv____,type,
    aTP_Lamp_alv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alw____,type,
    aTP_Lamp_alw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alx____,type,
    aTP_Lamp_alx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aly____,type,
    aTP_Lamp_aly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alz____,type,
    aTP_Lamp_alz: 
      !>[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__ama____,type,
    aTP_Lamp_ama: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amb____,type,
    aTP_Lamp_amb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amc____,type,
    aTP_Lamp_amc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amd____,type,
    aTP_Lamp_amd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ame____,type,
    aTP_Lamp_ame: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amf____,type,
    aTP_Lamp_amf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amg____,type,
    aTP_Lamp_amg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amh____,type,
    aTP_Lamp_amh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ami____,type,
    aTP_Lamp_ami: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amj____,type,
    aTP_Lamp_amj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amk____,type,
    aTP_Lamp_amk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aml____,type,
    aTP_Lamp_aml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amm____,type,
    aTP_Lamp_amm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amn____,type,
    aTP_Lamp_amn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amo____,type,
    aTP_Lamp_amo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amp____,type,
    aTP_Lamp_amp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amq____,type,
    aTP_Lamp_amq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amr____,type,
    aTP_Lamp_amr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ams____,type,
    aTP_Lamp_ams: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amt____,type,
    aTP_Lamp_amt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amu____,type,
    aTP_Lamp_amu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amv____,type,
    aTP_Lamp_amv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amw____,type,
    aTP_Lamp_amw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amx____,type,
    aTP_Lamp_amx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amy____,type,
    aTP_Lamp_amy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amz____,type,
    aTP_Lamp_amz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ana____,type,
    aTP_Lamp_ana: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anb____,type,
    aTP_Lamp_anb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anc____,type,
    aTP_Lamp_anc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__and____,type,
    aTP_Lamp_and: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ane____,type,
    aTP_Lamp_ane: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anf____,type,
    aTP_Lamp_anf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ang____,type,
    aTP_Lamp_ang: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anh____,type,
    aTP_Lamp_anh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ani____,type,
    aTP_Lamp_ani: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anj____,type,
    aTP_Lamp_anj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ank____,type,
    aTP_Lamp_ank: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anl____,type,
    aTP_Lamp_anl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anm____,type,
    aTP_Lamp_anm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ann____,type,
    aTP_Lamp_ann: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ano____,type,
    aTP_Lamp_ano: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anp____,type,
    aTP_Lamp_anp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anq____,type,
    aTP_Lamp_anq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anr____,type,
    aTP_Lamp_anr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ans____,type,
    aTP_Lamp_ans: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ant____,type,
    aTP_Lamp_ant: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anu____,type,
    aTP_Lamp_anu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anv____,type,
    aTP_Lamp_anv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anw____,type,
    aTP_Lamp_anw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anx____,type,
    aTP_Lamp_anx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__any____,type,
    aTP_Lamp_any: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anz____,type,
    aTP_Lamp_anz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aoa____,type,
    aTP_Lamp_aoa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aob____,type,
    aTP_Lamp_aob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoc____,type,
    aTP_Lamp_aoc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aod____,type,
    aTP_Lamp_aod: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoe____,type,
    aTP_Lamp_aoe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aof____,type,
    aTP_Lamp_aof: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aog____,type,
    aTP_Lamp_aog: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoh____,type,
    aTP_Lamp_aoh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoi____,type,
    aTP_Lamp_aoi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoj____,type,
    aTP_Lamp_aoj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aok____,type,
    aTP_Lamp_aok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aol____,type,
    aTP_Lamp_aol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aom____,type,
    aTP_Lamp_aom: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aon____,type,
    aTP_Lamp_aon: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoo____,type,
    aTP_Lamp_aoo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aop____,type,
    aTP_Lamp_aop: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoq____,type,
    aTP_Lamp_aoq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aor____,type,
    aTP_Lamp_aor: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aos____,type,
    aTP_Lamp_aos: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aot____,type,
    aTP_Lamp_aot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aou____,type,
    aTP_Lamp_aou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aov____,type,
    aTP_Lamp_aov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aow____,type,
    aTP_Lamp_aow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aox____,type,
    aTP_Lamp_aox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoy____,type,
    aTP_Lamp_aoy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoz____,type,
    aTP_Lamp_aoz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__apa____,type,
    aTP_Lamp_apa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apb____,type,
    aTP_Lamp_apb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apc____,type,
    aTP_Lamp_apc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apd____,type,
    aTP_Lamp_apd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ape____,type,
    aTP_Lamp_ape: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apf____,type,
    aTP_Lamp_apf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apg____,type,
    aTP_Lamp_apg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aph____,type,
    aTP_Lamp_aph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__api____,type,
    aTP_Lamp_api: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apj____,type,
    aTP_Lamp_apj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apk____,type,
    aTP_Lamp_apk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apl____,type,
    aTP_Lamp_apl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apm____,type,
    aTP_Lamp_apm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apn____,type,
    aTP_Lamp_apn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apo____,type,
    aTP_Lamp_apo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__app____,type,
    aTP_Lamp_app: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apq____,type,
    aTP_Lamp_apq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apr____,type,
    aTP_Lamp_apr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aps____,type,
    aTP_Lamp_aps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apt____,type,
    aTP_Lamp_apt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apu____,type,
    aTP_Lamp_apu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apv____,type,
    aTP_Lamp_apv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apw____,type,
    aTP_Lamp_apw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apx____,type,
    aTP_Lamp_apx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apy____,type,
    aTP_Lamp_apy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apz____,type,
    aTP_Lamp_apz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aqa____,type,
    aTP_Lamp_aqa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqb____,type,
    aTP_Lamp_aqb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqc____,type,
    aTP_Lamp_aqc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqd____,type,
    aTP_Lamp_aqd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqe____,type,
    aTP_Lamp_aqe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqf____,type,
    aTP_Lamp_aqf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqg____,type,
    aTP_Lamp_aqg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqh____,type,
    aTP_Lamp_aqh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqi____,type,
    aTP_Lamp_aqi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqj____,type,
    aTP_Lamp_aqj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqk____,type,
    aTP_Lamp_aqk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aql____,type,
    aTP_Lamp_aql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqm____,type,
    aTP_Lamp_aqm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqn____,type,
    aTP_Lamp_aqn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqo____,type,
    aTP_Lamp_aqo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqp____,type,
    aTP_Lamp_aqp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqq____,type,
    aTP_Lamp_aqq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqr____,type,
    aTP_Lamp_aqr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqs____,type,
    aTP_Lamp_aqs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqt____,type,
    aTP_Lamp_aqt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqu____,type,
    aTP_Lamp_aqu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqv____,type,
    aTP_Lamp_aqv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqw____,type,
    aTP_Lamp_aqw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqx____,type,
    aTP_Lamp_aqx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqy____,type,
    aTP_Lamp_aqy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqz____,type,
    aTP_Lamp_aqz: 
      !>[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__ara____,type,
    aTP_Lamp_ara: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arb____,type,
    aTP_Lamp_arb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arc____,type,
    aTP_Lamp_arc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ard____,type,
    aTP_Lamp_ard: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__are____,type,
    aTP_Lamp_are: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arf____,type,
    aTP_Lamp_arf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arg____,type,
    aTP_Lamp_arg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arh____,type,
    aTP_Lamp_arh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ari____,type,
    aTP_Lamp_ari: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arj____,type,
    aTP_Lamp_arj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ark____,type,
    aTP_Lamp_ark: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arl____,type,
    aTP_Lamp_arl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arm____,type,
    aTP_Lamp_arm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arn____,type,
    aTP_Lamp_arn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aro____,type,
    aTP_Lamp_aro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arp____,type,
    aTP_Lamp_arp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arq____,type,
    aTP_Lamp_arq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arr____,type,
    aTP_Lamp_arr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ars____,type,
    aTP_Lamp_ars: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__art____,type,
    aTP_Lamp_art: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aru____,type,
    aTP_Lamp_aru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arv____,type,
    aTP_Lamp_arv: 
      !>[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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : fun(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_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),$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 * A ) > A ) ).

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

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$o) ) ).

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

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__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_Ointeger__of__num,type,
    code_integer_of_num: fun(num,code_integer) ).

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

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

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

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun(num,code_integer) ).

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

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

tff(sy_c_Code__Target__Nat_Oint__of__nat,type,
    code_T6385005292777649522of_nat: fun(nat,int) ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

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

tff(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

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

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

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

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

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

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

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

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

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

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

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

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

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

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

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

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

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] : fun(set(A),filter(A)) ).

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

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

tff(sy_c_Finite__Set_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_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(A,B) ) > fun(A,C) ) ).

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_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__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_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_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

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

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,$o) > 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_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).

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

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

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

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

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

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

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

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

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > 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_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_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_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_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

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

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

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

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

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

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

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

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

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

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

tff(sy_c_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_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun(nat,num) ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

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

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : ( 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_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : 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_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_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

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

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

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

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

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

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

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_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] : fun(real,A) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

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

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

tff(sy_c_Set_OBex,type,
    bex: 
      !>[A: $tType] : fun(set(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

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

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

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : 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_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_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_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_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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_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_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__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: fun(vEBT_VEBT,fun(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__Intf__Functional_Oexperiment7207664_Otest,type,
    vEBT_Intf_test: ( nat * list(nat) * list(nat) ) > list(nat) ).

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_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: ( option(nat) * option(nat) ) > option(nat) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: ( option(nat) * option(nat) ) > option(nat) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: ( option(nat) * option(nat) ) > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,$o) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),$o) ) ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_xs,type,
    xs: list(nat) ).

tff(sy_v_ys,type,
    ys: list(nat) ).

% Relevant facts (9321)
tff(fact_0_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_1_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_2_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$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,$o),ord_less(nat),zero_zero(nat)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_4_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_5_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,A2),Na) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% power_eq_0_iff
tff(fact_6_nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,power_power(nat,X),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
        | ( Na = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_7_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_8_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$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),X),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_9_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$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,$o),ord_less(nat),zero_zero(nat)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_10_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_11_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_12_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,power_power(A,A2),Na) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_13_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na)) ) ) ).

% zero_le_power
tff(fact_14_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$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),Na)),aa(nat,A,power_power(A,B2),Na)) ) ) ) ).

% power_mono
tff(fact_15_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na)) ) ) ).

% zero_less_power
tff(fact_16_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,I),M)),aa(nat,nat,power_power(nat,I),Na))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_17_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na))
         => ( aa(A,$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_18_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,zero_zero(A)),Na) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_19_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_20_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),Na) = aa(nat,A,power_power(A,B2),Na) )
         => ( aa(A,$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)),Na)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_21_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,power_power(A,A2),Na) = aa(nat,A,power_power(A,B2),Na) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_22_less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% less_exp
tff(fact_23_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_24_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,power_power(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_25_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_26_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$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),X),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_27_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_28_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xsa) = aa(list(B),list(A),map(B,A,G),Xsa) )
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xsa))
         => ( aa(B,A,F2,X2) = aa(B,A,G,X2) ) ) ) ).

% map_eq_conv
tff(fact_29_le0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% le0
tff(fact_30_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_31_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_32_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% neq0_conv
tff(fact_33_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_34_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_35_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Na)) ).

% semiring_norm(76)
tff(fact_36_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_37_semiring__norm_I83_J,axiom,
    ! [Na: num] : one2 != bit0(Na) ).

% semiring_norm(83)
tff(fact_38_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_39_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ) ).

% numeral_less_iff
tff(fact_40_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,Na: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( M = Na ) ) ) ).

% numeral_eq_iff
tff(fact_41_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit0(M) = bit0(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_42_map__ident,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_ab(A,A)),X3) = X3 ).

% map_ident
tff(fact_43_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(71)
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,A2),aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
        <=> aa(A,$o,Q,X4) )
     => ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F2,X4) = aa(A,B,G,X4)
     => ( F2 = G ) ) ).

% ext
tff(fact_48_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(78)
tff(fact_49_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Na) ).

% semiring_norm(68)
tff(fact_50_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),one2) ).

% semiring_norm(75)
tff(fact_51_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ) ).

% numeral_le_iff
tff(fact_52_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_53_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_54_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),X),one2)
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_55_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ).

% nat_neq_iff
tff(fact_56_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_not_refl
tff(fact_57_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_58_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),T2)
     => ( S != T2 ) ) ).

% less_not_refl3
tff(fact_59_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_irrefl_nat
tff(fact_60_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% nat_less_induct
tff(fact_61_infinite__descent,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
              & ~ aa(nat,$o,P,M2) ) )
     => aa(nat,$o,P,Na) ) ).

% infinite_descent
tff(fact_62_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_63_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),X: A] :
      ( ! [X4: A] :
          ( ~ aa(A,$o,P,X4)
         => ? [Y2: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X4))
              & ~ aa(A,$o,P,Y2) ) )
     => aa(A,$o,P,X) ) ).

% infinite_descent_measure
tff(fact_64_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),Na) ).

% le_refl
tff(fact_65_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_66_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% eq_imp_le
tff(fact_67_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_68_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% nat_le_linear
tff(fact_69_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,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X4: nat] :
            ( aa(nat,$o,P,X4)
            & ! [Y2: nat] :
                ( aa(nat,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_70_subset__code_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),B3)
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => aa(set(A),$o,member(A,X2),B3) ) ) ).

% subset_code(1)
tff(fact_71_list_Omap__ident,axiom,
    ! [A: $tType,T2: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_ab(A,A)),T2) = T2 ).

% list.map_ident
tff(fact_72_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_73_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_74_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Na) ) ).

% zero_neq_numeral
tff(fact_75_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_76_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% gr0I
tff(fact_77_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_78_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% not_less0
tff(fact_79_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_80_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( Na != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_81_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                  & ~ aa(nat,$o,P,M2) ) ) )
       => aa(nat,$o,P,Na) ) ) ).

% infinite_descent0
tff(fact_82_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),X: A] :
      ( ! [X4: A] :
          ( ( aa(A,nat,V,X4) = zero_zero(nat) )
         => aa(A,$o,P,X4) )
     => ( ! [X4: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X4))
           => ( ~ aa(A,$o,P,X4)
             => ? [Y2: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X4))
                  & ~ aa(A,$o,P,Y2) ) ) )
       => aa(A,$o,P,X) ) ) ).

% infinite_descent0_measure
tff(fact_83_realpow__pos__nth,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ( aa(nat,real,power_power(real,R),Na) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_84_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X4)
            & ( aa(nat,real,power_power(real,X4),Na) = 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),Na) = A2 ) )
               => ( Y2 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_85_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% less_eq_nat.simps(1)
tff(fact_86_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_87_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_88_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),zero_zero(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_89_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_90_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_imp_le_nat
tff(fact_91_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_92_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_or_eq_imp_le
tff(fact_93_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ( M != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% le_neq_implies_less
tff(fact_94_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_95_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z: A] :
            ( aa(set(A),$o,member(A,Z),aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z) = aa(A,B,G,Z) ) )
       => ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_96_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z: A] :
          ( aa(set(A),$o,member(A,Z),aa(list(A),set(A),set2(A),X))
         => ( aa(A,B,F2,Z) = aa(A,B,G,Z) ) )
     => ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,G),X) ) ) ).

% list.map_cong0
tff(fact_97_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Xa: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z: A,Za: A] :
          ( aa(set(A),$o,member(A,Z),aa(list(A),set(A),set2(A),X))
         => ( aa(set(A),$o,member(A,Za),aa(list(A),set(A),set2(A),Xa))
           => ( ( aa(A,B,F2,Z) = aa(A,B,Fa,Za) )
             => ( Z = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,Fa),Xa) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_98_map__ext,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xsa) = aa(list(A),list(B),map(A,B,G),Xsa) ) ) ).

% map_ext
tff(fact_99_map__idI,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,A)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => ( aa(A,A,F2,X4) = X4 ) )
     => ( aa(list(A),list(A),map(A,A,F2),Xsa) = Xsa ) ) ).

% map_idI
tff(fact_100_map__cong,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ysa: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xsa = Ysa )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Ysa))
           => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xsa) = aa(list(A),list(B),map(A,B,G),Ysa) ) ) ) ).

% map_cong
tff(fact_101_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ysa: list(B),F2: fun(A,B)] :
      ( ? [Xs: list(A)] : Ysa = aa(list(A),list(B),map(A,B,F2),Xs)
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Ysa))
         => ? [Xa2: A] : X2 = aa(A,B,F2,Xa2) ) ) ).

% ex_map_conv
tff(fact_102_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_103_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_104_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_1: nat] : aa(nat,$o,P,X_1)
     => ( ! [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_105_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_le_numeral
tff(fact_106_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_107_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_less_numeral
tff(fact_108_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_109_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_110_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(1)
tff(fact_111_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(2)
tff(fact_112_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_113_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),zero_zero(A))
        <=> ( Na = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_114_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_115_insertCI,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( ( ~ aa(set(A),$o,member(A,A2),B3)
       => ( A2 = B2 ) )
     => aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) ) ).

% insertCI
tff(fact_116_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A3))
    <=> ( ( A2 = B2 )
        | aa(set(A),$o,member(A,A2),A3) ) ) ).

% insert_iff
tff(fact_117_insert__absorb2,axiom,
    ! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3) ).

% insert_absorb2
tff(fact_118_insert__subset,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),B3)
    <=> ( aa(set(A),$o,member(A,X),B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% insert_subset
tff(fact_119_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X4) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_120_Greatest__equality,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),Y3),X) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_121_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( bit0(X22) = bit0(Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_122_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_123_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X) ) ).

% order_refl
tff(fact_124_subsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(A),$o,member(A,X4),B3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% subsetI
tff(fact_125_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_126_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_127_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
    <=> ( Na != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_128_in__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,member(A,X),A3)
       => aa(set(A),$o,member(A,X),B3) ) ) ).

% in_mono
tff(fact_129_subsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,member(A,C2),A3)
       => aa(set(A),$o,member(A,C2),B3) ) ) ).

% subsetD
tff(fact_130_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_131_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_132_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(set(A),$o,member(A,X2),B3) ) ) ).

% subset_eq
tff(fact_133_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_134_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_135_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_136_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [T3: A] :
          ( aa(set(A),$o,member(A,T3),A3)
         => aa(set(A),$o,member(A,T3),B3) ) ) ).

% subset_iff
tff(fact_137_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_138_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(A,$o,Q,X4) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ) ).

% Collect_mono
tff(fact_139_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$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),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3) ) ) ).

% subset_trans
tff(fact_140_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_141_i0__lb,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Na) ).

% i0_lb
tff(fact_142_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)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).

% Collect_subset
tff(fact_143_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,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),B3)) ) ).

% less_eq_set_def
tff(fact_144_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),zero_zero(extended_enat))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_145_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q))
    <=> ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(A,$o,Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_146_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_147_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_148_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% psubset_subset_trans
tff(fact_149_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_150_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$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),$o),ord_less(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% subset_psubset_trans
tff(fact_151_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_152_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M2),N)
             => aa(extended_enat,$o,P,M2) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Na) ) ).

% enat_less_induct
tff(fact_153_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_154_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_155_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_156_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_le_cases
tff(fact_157_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,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_158_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,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_159_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_linear
tff(fact_160_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_161_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_eq_refl
tff(fact_162_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,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_163_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,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_164_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_165_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_166_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_167_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$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,X)),aa(A,B,G,X)) ) ) ).

% le_funE
tff(fact_168_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$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,X)),aa(A,B,G,X)) ) ) ).

% le_funD
tff(fact_169_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_170_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_171_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_172_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_173_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_174_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).

% order_trans
tff(fact_175_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_176_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_177_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_178_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_179_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% order_class.order_eq_iff
tff(fact_180_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_181_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_182_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_183_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_imp_not_less
tff(fact_184_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_185_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_186_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          | ( X = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_less_linear
tff(fact_187_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_188_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_not_sym
tff(fact_189_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,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_190_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,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_191_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X) ) ).

% order_less_irrefl
tff(fact_192_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,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_193_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,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_194_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_less_trans
tff(fact_195_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_196_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neq_iff
tff(fact_197_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_asym
tff(fact_198_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neqE
tff(fact_199_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_200_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_201_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_202_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_203_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_204_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_205_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N2: A] :
              ( aa(A,$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_206_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_207_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_208_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ( X != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_cases
tff(fact_209_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_210_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X4)
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_211_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_212_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_213_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_214_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_215_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ? [Z: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% dense
tff(fact_216_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_13) ) ).

% gt_ex
tff(fact_217_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ).

% lt_ex
tff(fact_218_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_219_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)),insert(A),B2),B3)) ) ).

% subset_insertI2
tff(fact_220_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)),insert(A),A2),B3)) ).

% subset_insertI
tff(fact_221_subset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),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)),insert(A),X),B3))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% subset_insert
tff(fact_222_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),D)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),D)) ) ).

% insert_mono
tff(fact_223_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ? [B5: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5) )
          & ~ aa(set(A),$o,member(A,A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_224_insert__commute,axiom,
    ! [A: $tType,X: A,Y: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) ).

% insert_commute
tff(fact_225_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A,B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,A2),A3)
     => ( ~ aa(set(A),$o,member(A,B2),B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3) )
        <=> $ite(
              A2 = B2,
              A3 = B3,
              ? [C4: set(A)] :
                ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),C4) )
                & ~ aa(set(A),$o,member(A,B2),C4)
                & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C4) )
                & ~ aa(set(A),$o,member(A,A2),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_226_insert__absorb,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = A3 ) ) ).

% insert_absorb
tff(fact_227_insert__ident,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),A3)
     => ( ~ aa(set(A),$o,member(A,X),B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% insert_ident
tff(fact_228_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,X),A3)
     => ~ ! [B5: set(A)] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5) )
           => aa(set(A),$o,member(A,X),B5) ) ) ).

% Set.set_insert
tff(fact_229_insertI2,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( aa(set(A),$o,member(A,A2),B3)
     => aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) ) ).

% insertI2
tff(fact_230_insertI1,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) ).

% insertI1
tff(fact_231_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A3))
     => ( ( A2 != B2 )
       => aa(set(A),$o,member(A,A2),A3) ) ) ).

% insertE
tff(fact_232_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_233_insert__compr,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_af(A,fun(set(A),fun(A,$o)),A2),B3)) ).

% insert_compr
tff(fact_234_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ).

% zero_le
tff(fact_235_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_236_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na)
         => ( Na != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_237_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Na),zero_zero(A)) ) ).

% not_less_zero
tff(fact_238_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ( Na != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na) ) ) ).

% gr_zeroI
tff(fact_239_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_240_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_le_less_linear
tff(fact_241_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,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_242_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,$o),ord_less_eq(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_243_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,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_244_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,$o),ord_less(B),B2),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_245_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_less_le_trans
tff(fact_246_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_le_less_trans
tff(fact_247_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_248_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_249_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_less_imp_le
tff(fact_250_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_not_less
tff(fact_251_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_not_le
tff(fact_252_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_253_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% dense_le_bounded
tff(fact_262_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),X)
                 => aa(A,$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),Z2) ) ) ) ).

% dense_ge_bounded
tff(fact_263_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_264_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_265_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_266_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_267_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_268_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% not_le_imp_less
tff(fact_269_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% less_le_not_le
tff(fact_270_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z2: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_le
tff(fact_271_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Y: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_ge
tff(fact_272_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_273_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_274_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_275_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% leI
tff(fact_276_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% leD
tff(fact_277_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A5: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A5)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_278_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_279_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_280_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),X) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_281_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_282_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),X) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_283_insert__subsetI,axiom,
    ! [A: $tType,X: A,A3: set(A),X5: set(A)] :
      ( aa(set(A),$o,member(A,X),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)),insert(A),X),X5)),A3) ) ) ).

% insert_subsetI
tff(fact_284_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) )
       => ? [X4: A] :
            ( aa(A,$o,P,X4)
            & ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X4)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_285_nat__descend__induct,axiom,
    ! [Na: nat,P: fun(nat,$o),M: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K2)
         => aa(nat,$o,P,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
           => ( ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I3)
                 => aa(nat,$o,P,I3) )
             => aa(nat,$o,P,K2) ) )
       => aa(nat,$o,P,M) ) ) ).

% nat_descend_induct
tff(fact_286_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_287_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_288_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)
             => ? [C5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),B2)
                  & ! [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),C5) )
                     => aa(A,$o,P,X3) )
                  & ! [D2: A] :
                      ( ! [X4: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D2) )
                         => aa(A,$o,P,X4) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C5) ) ) ) ) ) ) ).

% complete_interval
tff(fact_289_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),T2) ) ) ).

% pinf(6)
tff(fact_290_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X3) ) ) ).

% pinf(8)
tff(fact_291_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( M = Na ) ) ) ).

% of_nat_eq_iff
tff(fact_292_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(A,$o,Q,X4) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_293_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_294_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( zero_zero(nat) = Na ) ) ) ).

% of_nat_0_eq_iff
tff(fact_295_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_296_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% of_nat_numeral
tff(fact_297_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_iff
tff(fact_298_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% of_nat_le_iff
tff(fact_299_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,M),Na)) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),M)),Na) ) ).

% of_nat_power
tff(fact_300_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( ( aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,power_power(nat,B2),W2) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_301_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2) )
        <=> ( X = aa(nat,nat,power_power(nat,B2),W2) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_302_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_303_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% of_nat_0_less_iff
tff(fact_304_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,Na: nat,Y: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_305_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,X: num,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) )
        <=> ( Y = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_306_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),W2)),X) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_307_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,B2),W2)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_308_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),W2)),X) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_309_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,B2),W2)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_310_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),X)),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_311_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(nat,A,power_power(A,A2),Na))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_312_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_313_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Na)),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_314_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_315_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na) ) ) ).

% even_of_nat
tff(fact_316_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_317_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,member(A,C2),A3)
       => aa(set(A),$o,member(A,C2),B3) ) ) ).

% psubsetD
tff(fact_318_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,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),B3)) ) ).

% less_set_def
tff(fact_319_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% psubset_trans
tff(fact_320_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),X)
      | ( A2 = X )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),A2) ) ).

% verit_la_generic
tff(fact_321_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),X: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,X)
       => aa(A,$o,Q,X) ) ) ).

% predicate1D
tff(fact_322_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_323_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P2: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_324_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P2: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_325_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,X)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,X) ) ) ).

% rev_predicate1D
tff(fact_326_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( dvd_dvd(nat,M,Na)
     => ( dvd_dvd(nat,Na,M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_327_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,Na: nat] :
          ( dvd_dvd(A,X,Y)
         => dvd_dvd(A,aa(nat,A,power_power(A,X),Na),aa(nat,A,power_power(A,Y),Na)) ) ) ).

% dvd_power_same
tff(fact_328_int__ops_I3_J,axiom,
    ! [Na: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Na)) = aa(num,int,numeral_numeral(int),Na) ).

% int_ops(3)
tff(fact_329_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_330_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_331_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_332_nat__less__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Xa3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3)) ) ).

% nat_less_as_int
tff(fact_333_nat__leq__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3)) ) ).

% nat_leq_as_int
tff(fact_334_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_0_le_iff
tff(fact_335_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_336_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: num] : dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(num,A,numeral_numeral(A),bit0(Na))) ) ).

% even_numeral
tff(fact_337_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ).

% less_imp_of_nat_less
tff(fact_338_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_imp_less
tff(fact_339_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_340_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,Na: nat,M: nat] :
          ( dvd_dvd(A,X,Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => dvd_dvd(A,aa(nat,A,power_power(A,X),Na),aa(nat,A,power_power(A,Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_341_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat,B2: A,M: nat] :
          ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Na),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => dvd_dvd(A,aa(nat,A,power_power(A,A2),M),B2) ) ) ) ).

% power_le_dvd
tff(fact_342_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => dvd_dvd(A,aa(nat,A,power_power(A,A2),M),aa(nat,A,power_power(A,A2),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_343_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ~ dvd_dvd(nat,Na,M) ) ) ).

% nat_dvd_not_less
tff(fact_344_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_345_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( dvd_dvd(nat,K,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% dvd_imp_le
tff(fact_346_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_347_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X3) ) ) ).

% minf(7)
tff(fact_348_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),T2) ) ) ).

% minf(5)
tff(fact_349_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ( X3 != T2 ) ) ) ).

% minf(4)
tff(fact_350_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ( X3 != T2 ) ) ) ).

% minf(3)
tff(fact_351_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P2,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q2,X3) ) ) ) ) ) ) ).

% minf(2)
tff(fact_352_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P2,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q2,X3) ) ) ) ) ) ) ).

% minf(1)
tff(fact_353_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_354_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X3) ) ) ).

% pinf(7)
tff(fact_355_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),T2) ) ) ).

% pinf(5)
tff(fact_356_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ( X3 != T2 ) ) ) ).

% pinf(4)
tff(fact_357_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ( X3 != T2 ) ) ) ).

% pinf(3)
tff(fact_358_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P2,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q2,X3) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_359_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P2,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q2,X3) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_360_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_361_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X4)),aa(A,nat,M,Y2)) ) ) ) ).

% ex_has_least_nat
tff(fact_362_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(A,$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),Na)),aa(nat,A,power_power(A,B2),Na)) ) ) ) ).

% power_mono_odd
tff(fact_363_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% odd_pos
tff(fact_364_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Na))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_365_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_366_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Na)) ) ) ).

% zero_le_even_power
tff(fact_367_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)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_368_prop__restrict,axiom,
    ! [A: $tType,X: A,Z4: set(A),X5: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,X),Z4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,X) ) ) ).

% prop_restrict
tff(fact_369_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
              & ( A2 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_370_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( dvd_dvd(nat,aa(nat,nat,power_power(nat,K),M),aa(nat,nat,power_power(nat,K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_371_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Na)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            & ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
                & aa(A,$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)),Na)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_372_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X3) ) ) ).

% minf(8)
tff(fact_373_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),T2) ) ) ).

% minf(6)
tff(fact_374_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Na),aa(nat,A,power_power(A,B2),Na))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_375_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W2)),aa(nat,real,semiring_1_of_nat(real),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_376_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(num,real,numeral_numeral(real),W2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_377_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_378_int__eq__iff__numeral,axiom,
    ! [M: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_379_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Na)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_380_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : dvd_dvd(A,A2,zero_zero(A)) ) ).

% dvd_0_right
tff(fact_381_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_382_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> dvd_dvd(nat,M,Na) ) ) ).

% of_nat_dvd_iff
tff(fact_383_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( dvd_dvd(nat,M,Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_384_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_385_int__dvd__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> dvd_dvd(nat,M,Na) ) ).

% int_dvd_int_iff
tff(fact_386_complete__real,axiom,
    ! [S2: set(real)] :
      ( ? [X3: real] : aa(set(real),$o,member(real,X3),S2)
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa(set(real),$o,member(real,X4),S2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Z3) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( aa(set(real),$o,member(real,X3),S2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Y3) )
            & ! [Z3: real] :
                ( ! [X4: real] :
                    ( aa(set(real),$o,member(real,X4),S2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Z3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Z3) ) ) ) ) ).

% complete_real
tff(fact_387_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_388_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_389_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),Na) )
    <=> ( M = Na ) ) ).

% int_int_eq
tff(fact_390_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Na)
       => ~ dvd_dvd(int,Na,M) ) ) ).

% zdvd_not_zless
tff(fact_391_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => ( dvd_dvd(int,M,Na)
         => ( dvd_dvd(int,Na,M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_392_zdvd__imp__le,axiom,
    ! [Z2: int,Na: int] :
      ( dvd_dvd(int,Z2,Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),Na) ) ) ).

% zdvd_imp_le
tff(fact_393_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_394_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_395_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,A2,A2) ) ).

% dvd_refl
tff(fact_396_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_397_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_398_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_399_gcd__nat_Orefl,axiom,
    ! [A2: nat] : dvd_dvd(nat,A2,A2) ).

% gcd_nat.refl
tff(fact_400_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_401_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_402_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( dvd_dvd(nat,A2,A2)
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( dvd_dvd(nat,zero_zero(nat),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_416_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_417_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( dvd_dvd(nat,zero_zero(nat),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_418_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( dvd_dvd(nat,zero_zero(nat),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_419_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : dvd_dvd(nat,A2,zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_420_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% zle_int
tff(fact_421_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_422_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_423_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_424_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ag(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ag(A,fun(A,$o),B2)))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% subset_divisors_dvd
tff(fact_425_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ag(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ag(A,fun(A,$o),B2)))
        <=> ( dvd_dvd(A,A2,B2)
            & ~ dvd_dvd(A,B2,A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_426_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_427_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_428_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se8732182000553998342ip_bit(A,M,A2))
        <=> ~ ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_429_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_430_pred__subset__eq,axiom,
    ! [A: $tType,R2: set(A),S2: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),R2)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),S2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R2),S2) ) ).

% pred_subset_eq
tff(fact_431_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: 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),X)),Na)),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),X)),Na)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_432_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,Na: 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),X)),Na))
    <=> 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),X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_433_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: 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),X)),Na)),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),X)),Na)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_434_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,Na: 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),X)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_435_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_le
tff(fact_436_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_437_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_438_of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W2: int,Z2: int] :
          ( ( aa(int,A,ring_1_of_int(A),W2) = aa(int,A,ring_1_of_int(A),Z2) )
        <=> ( W2 = Z2 ) ) ) ).

% of_int_eq_iff
tff(fact_439_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_440_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_441_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_442_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_443_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_444_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_445_nat__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Na)) = Na ).

% nat_int
tff(fact_446_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_447_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z2: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z2) )
        <=> ( Z2 = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_448_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z2: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z2) = zero_zero(A) )
        <=> ( Z2 = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_449_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_450_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z2: int,Na: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z2) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( Z2 = aa(num,int,numeral_numeral(int),Na) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_451_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z2) ) ) ).

% of_int_le_iff
tff(fact_452_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% of_int_less_iff
tff(fact_453_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_454_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% of_int_of_nat_eq
tff(fact_455_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int,Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,power_power(int,Z2),Na)) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Z2)),Na) ) ).

% of_int_power
tff(fact_456_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W2: nat,X: 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),X) )
        <=> ( aa(nat,int,power_power(int,B2),W2) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_457_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B2: int,W2: nat] :
          ( ( aa(int,A,ring_1_of_int(A),X) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2) )
        <=> ( X = aa(nat,int,power_power(int,B2),W2) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_458_nat__le__0,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),zero_zero(int))
     => ( aa(int,nat,nat2,Z2) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_459_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_460_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% zless_nat_conj
tff(fact_461_int__nat__eq,axiom,
    ! [Z2: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2),Z2,zero_zero(int)) ).

% int_nat_eq
tff(fact_462_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2) ) ) ).

% of_int_0_le_iff
tff(fact_463_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_464_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ) ).

% of_int_0_less_iff
tff(fact_465_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_466_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_le_numeral_iff
tff(fact_467_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Na)),Z2) ) ) ).

% of_int_numeral_le_iff
tff(fact_468_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_less_numeral_iff
tff(fact_469_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Na)),Z2) ) ) ).

% of_int_numeral_less_iff
tff(fact_470_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ).

% zero_less_nat_eq
tff(fact_471_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,Na: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_472_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,Na: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) )
        <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_473_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,B2),W2)),X) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_474_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,power_power(int,B2),W2)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_475_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,B2),W2)),X) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_476_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,B2),W2)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_477_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z2)) = aa(int,A,ring_1_of_int(A),Z2) ) ) ) ).

% of_nat_nat
tff(fact_478_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) )
    <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_479_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_480_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_481_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_482_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),Na) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% unset_bit_nat_def
tff(fact_483_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),K) ).

% unset_bit_less_eq
tff(fact_484_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)) ).

% set_bit_greater_eq
tff(fact_485_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_486_nat__numeral__as__int,axiom,
    ! [X3: num] : aa(num,nat,numeral_numeral(nat),X3) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X3)) ).

% nat_numeral_as_int
tff(fact_487_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_488_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ? [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_489_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ! [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_490_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
       => ( ( aa(int,nat,nat2,Z2) = aa(int,nat,nat2,Z5) )
        <=> ( Z2 = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_491_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% nat_mono_iff
tff(fact_492_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(int,nat,nat2,Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z2) ) ).

% zless_nat_eq_int_zless
tff(fact_493_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),Na)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_494_nat__0__le,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = Z2 ) ) ).

% nat_0_le
tff(fact_495_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z2 )
    <=> ( ( M = aa(int,nat,nat2,Z2) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2) ) ) ).

% int_eq_iff
tff(fact_496_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% of_int_nonneg
tff(fact_497_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% of_int_pos
tff(fact_498_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: 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,Z2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% nat_less_eq_zless
tff(fact_499_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),X) ) ) ).

% of_nat_less_of_int_iff
tff(fact_500_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_501_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_502_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: 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)),Z2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z2) ) ) ).

% nat_le_eq_zle
tff(fact_503_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,I))
    <=> ( ! [N2: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N2) )
           => aa(nat,$o,P,N2) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_504_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ) ).

% le_nat_iff
tff(fact_505_nat__power__eq,axiom,
    ! [Z2: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(int,nat,nat2,aa(nat,int,power_power(int,Z2),Na)) = aa(nat,nat,power_power(nat,aa(int,nat,nat2,Z2)),Na) ) ) ).

% nat_power_eq
tff(fact_506_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_507_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(int,A,ring_1_of_int(A),K))
        <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K) ) ) ).

% even_of_int_iff
tff(fact_508_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),M)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_509_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( dvd_dvd(nat,aa(int,nat,nat2,Z2),M)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2),dvd_dvd(int,Z2,aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_510_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_511_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_512_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_less
tff(fact_513_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(int,nat,nat2,K))
      <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K) ) ) ).

% even_nat_iff
tff(fact_514_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_515_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_516_log__of__power__le,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B2),Na))
     => ( 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)),M)
         => 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),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_517_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_518_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z)) ) ).

% ex_less_of_int
tff(fact_519_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),X) ) ).

% ex_of_int_less
tff(fact_520_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z)) ) ).

% ex_le_of_int
tff(fact_521_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_522_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_523_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_524_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_525_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,one_one(A)),Na) = one_one(A) ) ).

% power_one
tff(fact_526_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Na) = one_one(A) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_527_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_528_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_529_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z2: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z2) = one_one(A) )
        <=> ( Z2 = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_530_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_531_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_532_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( one2 = Na ) ) ) ).

% one_eq_numeral_iff
tff(fact_533_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( aa(num,A,numeral_numeral(A),Na) = one_one(A) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_534_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( ( aa(nat,A,power_power(A,A2),M) = aa(nat,A,power_power(A,A2),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_535_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_536_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_537_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_538_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_539_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_540_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_541_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2) ) ) ).

% of_int_1_le_iff
tff(fact_542_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ) ).

% of_int_1_less_iff
tff(fact_543_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_544_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_545_log__less__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( 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),X)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_546_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),X)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_547_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_548_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),X)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_549_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_550_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_551_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$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,$o),ord_less(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_552_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).

% power_increasing_iff
tff(fact_553_log__le__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( 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),X)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_554_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),X)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_555_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_556_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),X)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_557_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_558_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),one2) ) ) ).

% numeral_le_one_iff
tff(fact_559_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Na) ) ) ).

% one_less_numeral_iff
tff(fact_560_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$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,$o),ord_less_eq(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_561_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2))
        <=> ( ( Na = zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_562_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_of_int
tff(fact_563_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_564_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)) ) ).

% take_bit_of_nat
tff(fact_565_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_566_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_567_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_568_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_569_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,one_one(A),A2) ) ).

% one_dvd
tff(fact_570_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_571_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_572_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_573_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_574_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ).

% take_bit_nonnegative
tff(fact_575_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_576_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_577_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_578_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_579_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),bit_se8732182000553998342ip_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_580_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_581_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_582_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_583_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_584_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_585_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_586_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_le_numeral
tff(fact_587_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_588_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_589_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na)) ) ) ).

% one_le_power
tff(fact_590_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_591_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_592_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,power_power(real,X),N)) ) ).

% real_arch_pow
tff(fact_593_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_594_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),X_13) ) ).

% linordered_field_no_ub
tff(fact_595_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X3) ) ).

% linordered_field_no_lb
tff(fact_596_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,power_power(A,zero_zero(A)),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_597_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$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),Na)),aa(nat,A,power_power(A,A2),N3)) ) ) ) ).

% power_strict_increasing
tff(fact_598_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_599_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$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),Na)),aa(nat,A,power_power(A,A2),N3)) ) ) ) ).

% power_increasing
tff(fact_600_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Na: nat] :
          ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Na),one_one(A))
        <=> ( dvd_dvd(A,A2,one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_601_real__arch__pow__inv,axiom,
    ! [Y: real,X: 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),X),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_602_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$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),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_603_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$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),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_604_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),one_one(A)) ) ).

% odd_one
tff(fact_605_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_606_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_607_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,power_power(A,A2),Na)) ) ) ) ).

% self_le_power
tff(fact_608_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A2),Na)) ) ) ) ).

% one_less_power
tff(fact_609_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,Na: nat] :
          ( ( X != zero_zero(A) )
         => ( dvd_dvd(A,aa(nat,A,power_power(A,X),M),aa(nat,A,power_power(A,X),Na))
          <=> ( dvd_dvd(A,X,one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_610_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,X: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            | ( X = one_one(A) ) )
         => dvd_dvd(A,X,aa(nat,A,power_power(A,X),Na)) ) ) ).

% dvd_power
tff(fact_611_log__of__power__eq,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,power_power(real,B2),Na) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_612_less__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,B2),Na)),M)
     => ( 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),Na)),aa(real,real,log(B2),M)) ) ) ).

% less_log_of_power
tff(fact_613_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_less_exp
tff(fact_614_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = zero_zero(A) )
        <=> dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_615_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_616_le__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,B2),Na)),M)
     => ( 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),Na)),aa(real,real,log(B2),M)) ) ) ).

% le_log_of_power
tff(fact_617_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_618_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_619_log__of__power__less,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B2),Na))
     => ( 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)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_620_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_621_VEBT__internal_Otwo__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ) ).

% VEBT_internal.two_realpow_ge_two
tff(fact_622_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_623_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_624_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_625_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_626_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_627_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_628_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_629_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_630_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_631_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_632_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_633_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_634_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_635_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_636_diff__0__eq__0,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),Na) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_637_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_638_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_639_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_640_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Na)
     => ( aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_641_Diff__insert0,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),A3)
     => ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3)) = aa(set(A),set(A),minus_minus(set(A),A3),B3) ) ) ).

% Diff_insert0
tff(fact_642_insert__Diff1,axiom,
    ! [A: $tType,X: A,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,member(A,X),B3)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),B3) = aa(set(A),set(A),minus_minus(set(A),A3),B3) ) ) ).

% insert_Diff1
tff(fact_643_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_644_idiff__0__right,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,Na),zero_zero(extended_enat)) = Na ).

% idiff_0_right
tff(fact_645_idiff__0,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),Na) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_646_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_647_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_648_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_649_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_650_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_651_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_652_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_653_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_654_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_655_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_656_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_657_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_658_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_659_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_660_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W2: num,Z2: 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)),Z2)) = 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))),Z2) ) ).

% add_numeral_left
tff(fact_661_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% numeral_plus_numeral
tff(fact_662_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_663_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_664_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_665_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_666_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) )
        <=> ( M = Na ) ) ) ).

% neg_numeral_eq_iff
tff(fact_675_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_676_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,minus_minus(A,A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_677_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(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,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_left
tff(fact_678_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(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_679_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(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,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_right
tff(fact_680_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(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_681_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_682_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_683_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_684_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,minus_minus(A,B2),A2) ) ).

% minus_diff_eq
tff(fact_685_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_686_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_687_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_add
tff(fact_688_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,Na),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% zero_less_diff
tff(fact_689_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( dvd_dvd(A,aa(A,A,uminus_uminus(A),X),Y)
        <=> dvd_dvd(A,X,Y) ) ) ).

% minus_dvd_iff
tff(fact_690_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( dvd_dvd(A,X,aa(A,A,uminus_uminus(A),Y))
        <=> dvd_dvd(A,X,Y) ) ) ).

% dvd_minus_iff
tff(fact_691_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% diff_is_0_eq
tff(fact_692_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_693_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_694_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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),X)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_695_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_696_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z2)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% of_int_minus
tff(fact_697_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_698_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( dvd_dvd(nat,M,one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_699_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_700_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_701_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_702_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_703_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_704_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2)) ) ).

% take_bit_xor
tff(fact_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_719_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_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_728_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_729_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(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_730_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,minus_minus(A,A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_731_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,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_732_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_733_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_734_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_735_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))) ) ).

% add_neg_numeral_simps(3)
tff(fact_736_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,minus_minus(A,zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_737_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_738_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_739_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,minus_minus(A,B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_740_less__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),one_one(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_one
tff(fact_741_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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),X)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_742_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_743_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X) ) ) ).

% ln_gt_zero_iff
tff(fact_744_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_745_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),Z2)) = 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),Z2)) ) ).

% of_int_add
tff(fact_746_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_747_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,minus_minus(int,W2),Z2)) = aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% of_int_diff
tff(fact_748_nat__zminus__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_749_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_750_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_751_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_752_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_753_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ) ).

% numeral_plus_one
tff(fact_754_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% one_plus_numeral
tff(fact_755_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_756_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_757_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) )
        <=> ( Na = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_758_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_759_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_760_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_761_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ).

% ln_ge_zero_iff
tff(fact_762_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_763_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z2) ) ).

% zle_add1_eq_le
tff(fact_764_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),M) ) ) ).

% neg_numeral_le_iff
tff(fact_765_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Na),M) ) ) ).

% neg_numeral_less_iff
tff(fact_766_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),aa(int,int,minus_minus(int,Z2),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ).

% zle_diff1_eq
tff(fact_767_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_768_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_769_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_770_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_771_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_772_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_773_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_774_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_775_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_776_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),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) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_777_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_778_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,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_779_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A2),Na)) ) ) ) ).

% power_minus_odd
tff(fact_780_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na) = aa(nat,A,power_power(A,A2),Na) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_781_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,Na: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na) = 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),X))),Na) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_782_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,Na: 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),X))),Na) )
        <=> ( Y = aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_783_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V2)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_784_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_785_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),X))),Na)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_786_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$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),X))),Na)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_787_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),X))),Na)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_788_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$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),X))),Na)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_789_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_790_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,minus_minus(A,Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_791_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_792_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$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_793_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$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_794_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,minus_minus(A,B2),A2)),A2) = B2 ) ) ) ).

% diff_add
tff(fact_795_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,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_796_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,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,B2),A2))
          <=> aa(A,$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_797_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,minus_minus(A,B2),A2)) = aa(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_798_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,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,minus_minus(A,B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_799_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,minus_minus(A,B2),A2)),C2) = aa(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_800_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,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,minus_minus(A,B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_801_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,minus_minus(A,C2),aa(A,A,minus_minus(A,B2),A2)) = aa(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_802_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,minus_minus(A,B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_803_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,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_804_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_805_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_806_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_807_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_808_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_809_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,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_810_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$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_811_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$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_812_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_xor_eq
tff(fact_813_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D3: A] : aa(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),D3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),aa(A,A,minus_minus(A,C2),D3)) ) ).

% add_diff_add
tff(fact_814_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) ) ).

% minus_diff_minus
tff(fact_815_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_816_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_817_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_818_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_819_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_820_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_821_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,minus_minus(A,A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_822_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,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,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_823_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(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_824_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(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_825_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,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_826_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_827_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_828_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,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_829_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_830_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D3) )
         => ( ( A2 = B2 )
          <=> ( C2 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_831_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_832_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_833_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_834_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_835_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_836_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_837_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_838_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_839_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(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_840_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_841_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_842_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_843_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,minus_minus(A,A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_844_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_845_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_846_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_847_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_848_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_849_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_850_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_851_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_852_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_853_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,minus_minus(int,zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_854_odd__nonzero,axiom,
    ! [Z2: 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)),Z2)),Z2) != zero_zero(int) ).

% odd_nonzero
tff(fact_855_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_856_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_857_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_858_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2)
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_859_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_860_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_861_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_862_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),Na)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ) ).

% of_nat_diff
tff(fact_863_int__minus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Na),M)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Na)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_864_odd__less__0__iff,axiom,
    ! [Z2: 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)),Z2)),Z2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_865_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_866_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q3)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_867_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2)
     => 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))),Z2) ) ).

% zless_imp_add1_zle
tff(fact_868_add1__zle__eq,axiom,
    ! [W2: int,Z2: 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))),Z2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ).

% add1_zle_eq
tff(fact_869_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_870_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,minus_minus(A,A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_871_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D3)) ) ) ) ).

% diff_mono
tff(fact_872_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,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2)) ) ) ).

% diff_left_mono
tff(fact_873_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,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% diff_right_mono
tff(fact_874_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_875_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_876_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_877_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,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_878_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,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_879_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_880_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_881_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_882_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_883_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_884_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_885_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),Na) ) ).

% neg_numeral_neq_numeral
tff(fact_886_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_neq_neg_numeral
tff(fact_887_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_888_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_889_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_890_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_891_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_892_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_893_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_894_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_mono
tff(fact_895_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_896_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)
         => ~ ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) ) ) ).

% less_eqE
tff(fact_897_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_898_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)
        <=> ? [C6: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C6) ) ) ).

% le_iff_add
tff(fact_899_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_900_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_901_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_902_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_903_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_904_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_905_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_strict_mono
tff(fact_906_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_907_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_908_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_909_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_910_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,minus_minus(A,C2),B2))
        <=> dvd_dvd(A,A2,aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_911_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A,Z2: A] :
          ( dvd_dvd(A,X,Y)
         => ( dvd_dvd(A,X,Z2)
           => dvd_dvd(A,X,aa(A,A,minus_minus(A,Y),Z2)) ) ) ) ).

% dvd_diff
tff(fact_912_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_913_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,Na),M) = zero_zero(nat) )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_914_double__diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$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),C3)
       => ( aa(set(A),set(A),minus_minus(set(A),B3),aa(set(A),set(A),minus_minus(set(A),C3),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_915_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),minus_minus(set(A),A3),B3)),A3) ).

% Diff_subset
tff(fact_916_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),D: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D),B3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),aa(set(A),set(A),minus_minus(set(A),C3),D)) ) ) ).

% Diff_mono
tff(fact_917_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_918_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_919_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_920_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_921_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_922_insert__Diff__if,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),B3) = $ite(aa(set(A),$o,member(A,X),B3),aa(set(A),set(A),minus_minus(set(A),A3),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),minus_minus(set(A),A3),B3))) ).

% insert_Diff_if
tff(fact_923_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,minus_minus(int,K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_924_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_925_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% le_diff_iff
tff(fact_926_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_927_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Na),L)) ) ).

% diff_le_mono
tff(fact_928_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ).

% diff_le_self
tff(fact_929_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,minus_minus(nat,C2),A2)),aa(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_930_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_931_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_932_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_933_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_934_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))),aa(real,real,uminus_uminus(real),X)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_935_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M4: nat,N: nat] : Z2 != aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M4)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_936_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,K,M)
     => ( dvd_dvd(nat,K,Na)
       => dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_937_nat__minus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,minus_minus(nat,X3),Xa3) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_minus_as_int
tff(fact_938_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N: nat] : Z2 != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% int_cases2
tff(fact_939_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_940_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ? [B4: A] : aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_941_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( dvd_dvd(int,K,aa(int,int,minus_minus(int,M),Na))
     => ( dvd_dvd(int,K,Na)
       => dvd_dvd(int,K,M) ) ) ).

% zdvd_zdiffD
tff(fact_942_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,T2: int] :
      ( dvd_dvd(int,D3,T2)
    <=> dvd_dvd(int,aa(int,int,uminus_uminus(int),D3),T2) ) ).

% uminus_dvd_conv(1)
tff(fact_943_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,T2: int] :
      ( dvd_dvd(int,D3,T2)
    <=> dvd_dvd(int,D3,aa(int,int,uminus_uminus(int),T2)) ) ).

% uminus_dvd_conv(2)
tff(fact_944_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_945_take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_946_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,minus_minus(real,X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_947_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_948_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_949_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X4: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X4)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),one_one(int))))
          & ! [Y2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y2)),X)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y2),one_one(int)))) )
             => ( Y2 = X4 ) ) ) ) ).

% floor_exists1
tff(fact_950_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))) ) ) ).

% floor_exists
tff(fact_951_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(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,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_952_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,X),Y)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_953_nat__diff__distrib,axiom,
    ! [Z5: int,Z2: 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),Z2)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,Z2),Z5)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_954_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => 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)),Z2)) ) ).

% le_imp_0_less
tff(fact_955_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) != aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ) ).

% take_bit_incr_eq
tff(fact_956_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,minus_minus(real,X),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_957_ln__less__self,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),X)),X) ) ).

% ln_less_self
tff(fact_958_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,minus_minus(int,K),L))
    <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_diff_iff
tff(fact_959_zdiff__int__split,axiom,
    ! [P: fun(int,$o),X: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,X),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_960_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( 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_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)) ) ) ).

% XOR_lower
tff(fact_961_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,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_962_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,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_963_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_neq_neg_numeral
tff(fact_964_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_le_numeral
tff(fact_965_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_le_neg_numeral
tff(fact_966_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_less_numeral
tff(fact_967_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_less_neg_numeral
tff(fact_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_978_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$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),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_979_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_980_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_981_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_982_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_983_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)
         => ~ ! [C5: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_984_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_985_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_986_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_987_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_988_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_989_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_990_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_neq_neg_numeral
tff(fact_991_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),Na) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_992_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_993_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_994_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit0(Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_Bit0
tff(fact_995_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_996_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(10)
tff(fact_997_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(9)
tff(fact_998_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(10)
tff(fact_999_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(9)
tff(fact_1000_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_1001_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A,C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),C3)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),C3))
        & ~ aa(set(A),$o,member(A,X),A3) ) ) ).

% subset_Diff_insert
tff(fact_1002_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1003_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,minus_minus(nat,A2),C2)),aa(nat,nat,minus_minus(nat,B2),C2)) ) ) ).

% diff_less_mono
tff(fact_1004_diff__nat__eq__if,axiom,
    ! [Z2: int,Z5: int] :
      aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z2)),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,Z2),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z2),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_1005_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,Na),M))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
        | dvd_dvd(nat,M,Na) ) ) ).

% dvd_minus_self
tff(fact_1006_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),Na))
     => ( dvd_dvd(nat,K,Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => dvd_dvd(nat,K,M) ) ) ) ).

% dvd_diffD
tff(fact_1007_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),Na))
     => ( dvd_dvd(nat,K,M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => dvd_dvd(nat,K,Na) ) ) ) ).

% dvd_diffD1
tff(fact_1008_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( dvd_dvd(nat,M,Na)
      <=> dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_1009_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_1010_zle__iff__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z2)
    <=> ? [N2: nat] : Z2 = 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_1011_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1012_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),bit0(Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_1013_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_1014_ln__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),X) ) ).

% ln_bound
tff(fact_1015_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1016_ln__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1017_ln__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)) ) ).

% ln_gt_zero
tff(fact_1018_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)) ) ).

% ln_ge_zero
tff(fact_1019_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_1020_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_le_neg_numeral
tff(fact_1021_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_1022_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_less_neg_numeral
tff(fact_1023_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_1024_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_1025_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_1026_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_1027_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_1028_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_1029_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_1030_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_1031_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_1032_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_1033_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_1034_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_1035_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_1036_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% field_le_epsilon
tff(fact_1044_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_1045_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_1046_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_1047_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != 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)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_1048_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_1049_negative__zle__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),zero_zero(int)) ).

% negative_zle_0
tff(fact_1050_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_1051_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_1052_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ).

% int_one_le_iff_zero_less
tff(fact_1053_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1054_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_1055_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_1056_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,aa(A,A,minus_minus(A,Y),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_1057_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,uminus_uminus(A),A2))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% even_minus
tff(fact_1058_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(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_xor_iff
tff(fact_1059_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_1060_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_1061_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_1062_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_1063_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,aa(nat,nat,power_power(nat,I),M),aa(nat,nat,power_power(nat,I),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_1064_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_1065_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_less_exp
tff(fact_1066_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_1067_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A2 = one_one(A) )
            | ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_1068_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,M: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1069_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] :
          aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na),aa(nat,A,power_power(A,A2),Na),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A2),Na))) ) ).

% uminus_power_if
tff(fact_1070_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,power_power(real,U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_1071_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_1072_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_1073_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_1074_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_1075_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : aa(A,$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,X),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_1076_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$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,X),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))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1077_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ aa(A,$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,X),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_1078_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$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,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1079_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_1080_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_1081_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% compl_less_compl_iff
tff(fact_1082_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% compl_le_compl_iff
tff(fact_1083_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_1084_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),Na) = semiri8178284476397505188at_aux(A,aTP_Lamp_ah(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_1085_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1086_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B4: real,C5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B4)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B4),C5)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),C5)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C5) ) ) ) )
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
               => ? [D2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
                    & ! [A4: real,B4: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,B4),A4)),D2) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B4) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1087_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),A3)
     => ( ~ aa(set(A),$o,member(A,C2),B3)
       => aa(set(A),$o,member(A,C2),aa(set(A),set(A),minus_minus(set(A),A3),B3)) ) ) ).

% DiffI
tff(fact_1088_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),minus_minus(set(A),A3),B3))
    <=> ( aa(set(A),$o,member(A,C2),A3)
        & ~ aa(set(A),$o,member(A,C2),B3) ) ) ).

% Diff_iff
tff(fact_1089_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B3)),B3) = aa(set(A),set(A),minus_minus(set(A),A3),B3) ).

% Diff_idemp
tff(fact_1090_ComplI,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( ~ aa(set(A),$o,member(A,C2),A3)
     => aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% ComplI
tff(fact_1091_Compl__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ~ aa(set(A),$o,member(A,C2),A3) ) ).

% Compl_iff
tff(fact_1092_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_1093_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_1094_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_1095_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_1096_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1097_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_1098_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_1099_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_1100_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_1101_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_1102_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_1103_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% div_minus_minus
tff(fact_1104_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1105_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2))
            <=> dvd_dvd(A,B2,C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1106_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_1107_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Na = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_1108_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V2))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit0
tff(fact_1109_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_1110_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_1111_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1112_real__add__minus__iff,axiom,
    ! [X: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( X = A2 ) ) ).

% real_add_minus_iff
tff(fact_1113_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(6)
tff(fact_1114_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1115_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1116_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1117_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1118_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_1119_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1120_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1121_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1122_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_1123_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A2) ) ).

% div_minus1_right
tff(fact_1124_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_1125_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1126_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1127_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_1128_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_1129_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% add_gr_0
tff(fact_1130_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1131_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1132_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1133_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)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1134_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)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1135_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = bit0(one2) ).

% semiring_norm(2)
tff(fact_1136_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_1137_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_1138_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_1139_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,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1140_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,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1141_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1142_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1143_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_1144_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_1145_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% diff_numeral_simps(3)
tff(fact_1146_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% diff_numeral_simps(2)
tff(fact_1147_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,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1148_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,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1149_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1150_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1151_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1152_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,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_1153_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_1154_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_1155_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_1156_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1157_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% diff_numeral_special(3)
tff(fact_1158_even__succ__div__two,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),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),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1159_odd__succ__div__two,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),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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1160_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)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1161_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,minus_minus(nat,M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ) ) ).

% even_diff_nat
tff(fact_1162_minus__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,minus_minus(real,X),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_1163_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),minus_minus(set(A),A3),B3))
     => ~ ( aa(set(A),$o,member(A,C2),A3)
         => aa(set(A),$o,member(A,C2),B3) ) ) ).

% DiffE
tff(fact_1164_ComplD,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A3))
     => ~ aa(set(A),$o,member(A,C2),A3) ) ).

% ComplD
tff(fact_1165_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),minus_minus(set(A),A3),B3))
     => aa(set(A),$o,member(A,C2),A3) ) ).

% DiffD1
tff(fact_1166_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),minus_minus(set(A),A3),B3))
     => ~ aa(set(A),$o,member(A,C2),B3) ) ).

% DiffD2
tff(fact_1167_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),B3))) ).

% minus_set_def
tff(fact_1168_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3))) ).

% uminus_set_def
tff(fact_1169_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_1170_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% Nat.diff_cancel
tff(fact_1171_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_cancel2
tff(fact_1172_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),K)),J) ).

% diff_commute
tff(fact_1173_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1174_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1175_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1176_real__of__int__div4,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),X))) ).

% real_of_int_div4
tff(fact_1177_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_1178_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_1179_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ak(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% set_diff_eq
tff(fact_1180_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_1181_real__of__int__div,axiom,
    ! [D3: int,Na: int] :
      ( dvd_dvd(int,D3,Na)
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),D3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),D3)) ) ) ).

% real_of_int_div
tff(fact_1182_real__of__int__div2,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),X)))) ).

% real_of_int_div2
tff(fact_1183_real__of__int__div3,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),X)))),one_one(real)) ).

% real_of_int_div3
tff(fact_1184_real__of__nat__div4,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),X))) ).

% real_of_nat_div4
tff(fact_1185_real__of__nat__div,axiom,
    ! [D3: nat,Na: nat] :
      ( dvd_dvd(nat,D3,Na)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),D3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),D3)) ) ) ).

% real_of_nat_div
tff(fact_1186_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% div_minus_right
tff(fact_1187_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na)) ) ).

% power_divide
tff(fact_1188_add__One__commute,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2) ).

% add_One_commute
tff(fact_1189_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)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1190_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( dvd_dvd(A,C2,A2)
           => ( dvd_dvd(A,C2,B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1191_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A2: A] :
          ( dvd_dvd(A,D3,B2)
         => ( dvd_dvd(A,B2,A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1192_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ).

% div_le_dividend
tff(fact_1193_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),K)) ) ).

% div_le_mono
tff(fact_1194_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_1195_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = M )
     => ( Na = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_1196_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Na) = Na ).

% plus_nat.add_0
tff(fact_1197_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1198_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% less_add_eq_less
tff(fact_1199_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_1200_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_1201_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_1202_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_1203_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_1204_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_1205_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_1206_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1207_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1208_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_1209_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_1210_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_1211_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_1212_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_1213_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_1214_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% add_leD2
tff(fact_1215_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% add_leD1
tff(fact_1216_le__add2,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% le_add2
tff(fact_1217_le__add1,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) ).

% le_add1
tff(fact_1218_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% add_leE
tff(fact_1219_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1220_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1221_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1222_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1223_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1224_nat__div__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_div_as_int
tff(fact_1225_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),Na) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( Na = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_1226_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),aa(int,int,aa(int,fun(int,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_1227_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),aa(int,int,aa(int,fun(int,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_1228_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1229_add__diff__assoc__enat,axiom,
    ! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z2),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z2)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),Y)),Z2) ) ) ).

% add_diff_assoc_enat
tff(fact_1230_real__of__nat__div2,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),X)))) ).

% real_of_nat_div2
tff(fact_1231_real__of__nat__div3,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),X)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_1232_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1233_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( 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),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1234_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),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),M),Na))) ) ).

% div_exp_eq
tff(fact_1235_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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_1236_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)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1237_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1238_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1239_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1240_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1241_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1242_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$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_1243_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1244_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1245_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1246_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1247_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$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_1248_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$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_1249_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1250_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_1251_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_1252_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_1253_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A))
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_1254_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1255_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_1256_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1257_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1258_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1259_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_one_over
tff(fact_1260_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1261_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),C2)
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1262_unit__div__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),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1263_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1264_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1265_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_1266_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_1267_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,Na: nat] :
          ( dvd_dvd(A,B2,A2)
         => ( aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na)) ) ) ) ).

% div_power
tff(fact_1268_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1269_ln__div,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_1270_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)),aa(int,int,aa(int,fun(int,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_1271_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)),aa(int,int,aa(int,fun(int,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_1272_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)),aa(int,int,aa(int,fun(int,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_1273_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1274_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1275_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1276_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)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_1277_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),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_1278_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: 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)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1279_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1280_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1281_zdiv__mono2,axiom,
    ! [A2: int,B6: 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)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6)) ) ) ) ).

% zdiv_mono2
tff(fact_1282_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),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2)) ) ) ).

% zdiv_mono1
tff(fact_1283_real__0__less__add__iff,axiom,
    ! [X: 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),X),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_less_add_iff
tff(fact_1284_real__add__less__0__iff,axiom,
    ! [X: 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),X),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_less_0_iff
tff(fact_1285_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),X)
     => ( 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),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1286_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1287_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_le_add_iff
tff(fact_1288_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_le_0_iff
tff(fact_1289_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1290_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_1291_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_1292_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M4)),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,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_1293_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Na: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ? [Y2: A] :
                ( aa(A,$o,P,Y2)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X4)) ) )
       => ? [Y3: A] :
            ( aa(A,$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)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1294_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1295_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_1296_int__plus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_1297_zadd__int__left,axiom,
    ! [M: nat,Na: nat,Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),Z2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))),Z2) ).

% zadd_int_left
tff(fact_1298_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1299_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1300_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1301_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1302_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$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_1303_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_1304_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_less
tff(fact_1305_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_le
tff(fact_1306_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$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_1307_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$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_1308_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1309_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1310_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1311_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_1312_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Na) ) ).

% numeral_Bit0_div_2
tff(fact_1313_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1314_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1315_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Na: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),Na)) ) ) ) ) ).

% power_diff
tff(fact_1316_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_1317_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% div_greater_zero_iff
tff(fact_1318_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_1319_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = M )
      <=> ( Na = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1320_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_1321_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_1322_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => 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)),X))),X) ) ).

% ln_add_one_self_le_self
tff(fact_1323_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => 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)),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_1324_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,Na: 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),Na))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1325_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real))) ) ).

% int_le_real_less
tff(fact_1326_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_1327_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% xor_nat_def
tff(fact_1328_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_1329_nat__plus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_plus_as_int
tff(fact_1330_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$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_1331_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$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_1332_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) = X ) ).

% field_sum_of_halves
tff(fact_1333_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: 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) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1334_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,M),Na)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1335_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_1336_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1337_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Na)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_1338_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_1339_log__base__change,axiom,
    ! [A2: real,B2: real,X: 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),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_1340_log__divide,axiom,
    ! [A2: real,X: 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)),X)
         => ( 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),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1341_nat__add__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( 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),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_1342_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)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1343_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_1344_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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_1345_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_1346_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( ( Na != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Na))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1347_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1348_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1349_nat__induct2,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$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,Na) ) ) ) ).

% nat_induct2
tff(fact_1350_minus__1__div__exp__eq__int,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_1351_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))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1352_log__base__pow,axiom,
    ! [A2: real,Na: nat,X: 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),Na)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_1353_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_le_swap2
tff(fact_1354_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_1355_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X)) ) ) ).

% compl_mono
tff(fact_1356_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_less_swap2
tff(fact_1357_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_1358_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,X),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_1359_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_1360_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_1361_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_1362_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A] : aa(A,A,artanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),aa(A,A,minus_minus(A,one_one(A)),X)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_1363_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_1364_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),zero_zero(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_1365_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_1366_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_1367_VEBT__internal_Opower__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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)) ) ) ).

% VEBT_internal.power_minus_is_div
tff(fact_1368_VEBT__internal_Opow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,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) ).

% VEBT_internal.pow_sum
tff(fact_1369_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),X)
        <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Y) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1370_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$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),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_1371_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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),Na)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Na)),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),Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1372_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Na)),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),Na),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),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1373_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),X)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_1374_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_1375_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_1376_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_1377_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_1378_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_1379_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% numeral_times_numeral
tff(fact_1380_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W2: num,Z2: 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)),Z2)) = 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))),Z2) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1381_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_1382_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_1383_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_1384_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_1385_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_1386_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_mult
tff(fact_1387_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z2)) = 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),Z2)) ) ).

% of_int_mult
tff(fact_1388_real__divide__square__eq,axiom,
    ! [R3: real,A2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R3),R3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R3) ).

% real_divide_square_eq
tff(fact_1389_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N2: int] : X = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_ceiling_cancel
tff(fact_1390_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_1391_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_1392_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_1393_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_1394_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1395_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_1396_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_1397_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1398_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1399_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_1400_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_1401_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_1402_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1403_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1404_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1405_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1406_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1407_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,minus_minus(A,B2),C2)) = aa(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_1408_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,minus_minus(A,A2),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(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_1409_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1410_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1411_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1412_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(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)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ).

% semiring_norm(170)
tff(fact_1413_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: 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(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ).

% semiring_norm(171)
tff(fact_1414_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = 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))),Y) ) ).

% semiring_norm(172)
tff(fact_1415_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_1416_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_1417_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_1418_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_1419_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z2) = aa(A,A,uminus_uminus(A),Z2) ) ).

% mult_minus1
tff(fact_1420_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z2) ) ).

% mult_minus1_right
tff(fact_1421_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_1422_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_1423_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_1424_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1425_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1426_not__real__square__gt__zero,axiom,
    ! [X: 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),X),X))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1427_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1428_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% ceiling_numeral
tff(fact_1429_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% ceiling_of_nat
tff(fact_1430_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_1431_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),Na)) = aa(num,int,numeral_numeral(int),Na) ) ).

% round_numeral
tff(fact_1432_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% round_of_nat
tff(fact_1433_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,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_1434_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),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_1435_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$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_1436_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,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$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_1437_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$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_1438_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,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$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_1439_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1440_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1441_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1442_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1443_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1444_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1445_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] : 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))),Na)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_1446_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat,A2: 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))),Na)),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))),Na)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_1447_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_1448_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1449_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),Na))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% power_add_numeral
tff(fact_1450_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),Na))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)))),B2) ) ).

% power_add_numeral2
tff(fact_1451_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_1452_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_1453_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$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_1454_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,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$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_1455_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$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_1456_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,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$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_1457_even__mult__iff,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),times_times(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_mult_iff
tff(fact_1458_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_1459_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_1460_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X) ) ) ).

% zero_less_ceiling
tff(fact_1461_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_1462_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V2)),X) ) ) ).

% numeral_less_ceiling
tff(fact_1463_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1464_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X) ) ) ).

% one_le_ceiling
tff(fact_1465_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1466_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ).

% one_less_ceiling
tff(fact_1467_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_1468_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_neg_numeral
tff(fact_1469_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_diff_numeral
tff(fact_1470_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,Na: nat] : archimedean_ceiling(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ).

% ceiling_numeral_power
tff(fact_1471_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)) ) ).

% round_neg_numeral
tff(fact_1472_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,X))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_1473_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_1474_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).

% zero_le_ceiling
tff(fact_1475_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_1476_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1477_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))),X) ) ) ).

% numeral_le_ceiling
tff(fact_1478_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_1479_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_1480_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X) ) ) ).

% neg_numeral_less_ceiling
tff(fact_1481_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_1482_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1483_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1484_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1485_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1486_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X)) ) ).

% ceiling_ge_round
tff(fact_1487_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_1488_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_1489_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_1490_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_1491_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_al(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_1492_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_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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,minus_minus(A,A2),B2)),C2) = aa(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_1507_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,minus_minus(A,B2),C2)) = aa(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_1508_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,minus_minus(A,B2),C2)),A2) = aa(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_1509_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,minus_minus(A,B2),C2)) = aa(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_1510_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D: A,Q: fun(A,$o)] :
          ( ! [X4: A,K2: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D))) )
         => ( ! [X4: A,K2: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D))) )
           => ! [X3: A,K4: A] :
                ( ( aa(A,$o,P,X3)
                  | aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D)))
                  | aa(A,$o,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1511_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D: A,Q: fun(A,$o)] :
          ( ! [X4: A,K2: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D))) )
         => ( ! [X4: A,K2: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D))) )
           => ! [X3: A,K4: A] :
                ( ( aa(A,$o,P,X3)
                  & aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D)))
                  & aa(A,$o,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1512_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_1513_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_1514_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),Na)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_commutes
tff(fact_1515_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na)) ) ).

% power_mult_distrib
tff(fact_1516_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,power_power(A,X),Na)) ) ) ) ).

% power_commuting_commutes
tff(fact_1517_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_1518_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_1519_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,C2,D3)
           => 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),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_1520_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_1521_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_1522_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_1523_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))
         => ? [B7: A,C7: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),C7) )
              & dvd_dvd(A,B7,B2)
              & dvd_dvd(A,C7,C2) ) ) ) ).

% division_decomp
tff(fact_1524_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_1525_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))
         => ~ ! [X4: A,Y3: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X4),Y3) )
               => ( dvd_dvd(A,X4,A2)
                 => ~ dvd_dvd(A,Y3,B2) ) ) ) ) ).

% dvd_productE
tff(fact_1526_dvd__def,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) ) ) ).

% dvd_def
tff(fact_1527_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_1528_dvdE,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) ) ) ).

% dvdE
tff(fact_1529_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1530_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_1531_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_am(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1532_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_an(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1533_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X)) ) ) ).

% ceiling_mono
tff(fact_1534_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ).

% le_of_int_ceiling
tff(fact_1535_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% ceiling_less_cancel
tff(fact_1536_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_1537_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_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_1558_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_1559_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_1560_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_1561_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_1562_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_1563_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_1564_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_1565_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_1566_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_1567_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_1568_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_1569_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_1570_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_1571_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_1572_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),Na)) ) ) ) ).

% less_1_mult
tff(fact_1573_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_1574_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_1575_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1576_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1577_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1578_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1579_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $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_1580_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> $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_1581_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1582_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W2: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% numeral_times_minus_swap
tff(fact_1583_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1584_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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)),D3) )
        <=> ( 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,minus_minus(A,B2),A2)),E2)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1585_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,minus_minus(A,X),Y)) ) ).

% square_diff_square_factored
tff(fact_1586_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),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),X),aa(A,A,minus_minus(A,Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1587_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1588_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),Na)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1589_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_1590_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_1591_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_1592_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_1593_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_1594_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_1595_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_1596_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D3: A,C2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( dvd_dvd(A,D3,C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1597_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_1598_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)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1599_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)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_1600_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_1601_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_1602_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% div_mult2_eq'
tff(fact_1603_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Na: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_add
tff(fact_1604_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)) ).

% real_minus_mult_self_le
tff(fact_1605_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$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,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)) ) ) ).

% ceiling_divide_upper
tff(fact_1606_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na)) = 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))),Na)) ) ).

% minus_power_mult_self
tff(fact_1607_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_1608_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,X)))) ).

% real_nat_ceiling_ge
tff(fact_1609_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R3),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R3)))) ) ).

% of_nat_ceiling
tff(fact_1610_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),A2) ) ) ).

% ceiling_le
tff(fact_1611_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% ceiling_le_iff
tff(fact_1612_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),X) ) ) ).

% less_ceiling_iff
tff(fact_1613_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_1614_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_1615_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_1616_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_1617_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_1618_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_1619_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1620_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_1621_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_1622_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1623_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_1624_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_1625_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_1626_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_1627_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_1628_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1629_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1630_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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),X)),X) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1631_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$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),X),Y)),X) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1632_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_1633_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_1634_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1635_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : aa(A,$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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1636_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1637_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ aa(A,$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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1638_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)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1639_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,aa(A,fun(A,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_1640_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),aa(A,A,aa(A,fun(A,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)),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_1641_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$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_1642_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,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$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_1643_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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$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_1644_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,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1645_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1646_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1647_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1648_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1649_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),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_1650_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) = aa(A,A,aa(A,fun(A,A),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_1651_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2)),D3) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1652_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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,minus_minus(A,B2),A2)),E2)),D3)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1653_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_add_eq_iff
tff(fact_1654_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% add_divide_eq_iff
tff(fact_1655_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1656_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1657_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),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),Z2)) ) ) ) ) ).

% add_frac_eq
tff(fact_1658_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = $ite(Z2 = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z2)),B2)),Z2)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1659_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z2)),B2) = $ite(Z2 = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1660_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2)),D3) ) ) ).

% less_add_iff1
tff(fact_1661_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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,minus_minus(A,B2),A2)),E2)),D3)) ) ) ).

% less_add_iff2
tff(fact_1662_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1663_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1664_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),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),Z2)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1665_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z2: A] :
          aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = $ite(Z2 = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z2)),B2)),Z2)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1666_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na))) ) ) ).

% power_gt1_lemma
tff(fact_1667_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Na))) ) ) ).

% power_less_power_Suc
tff(fact_1668_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_1669_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_1670_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] : aa(A,$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)),X)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1671_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1672_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1673_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1674_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> $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_1675_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $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_1676_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C5) ) ) ) ).

% unit_dvdE
tff(fact_1677_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_1678_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( dvd_dvd(A,A2,B2)
             => ( dvd_dvd(A,C2,D3)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D3),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),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1679_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,aa(A,A,aa(A,fun(A,A),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_1680_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,aa(A,A,aa(A,fun(A,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_1681_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)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1682_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na) = 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))),Na)),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_minus
tff(fact_1683_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D: A,T2: A] :
          ( dvd_dvd(A,D3,D)
         => ! [X3: A,K4: A] :
              ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2))
            <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D))),T2)) ) ) ) ).

% inf_period(4)
tff(fact_1684_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D: A,T2: A] :
          ( dvd_dvd(A,D3,D)
         => ! [X3: A,K4: A] :
              ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2))
            <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D))),T2)) ) ) ) ).

% inf_period(3)
tff(fact_1685_unit__eq__div1,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),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_1686_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_1687_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)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_1688_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_1689_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_1690_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))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1691_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ! [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)),X)) ) ).

% reals_Archimedean3
tff(fact_1692_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_1693_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] :
          aa(nat,A,power_power(A,Z2),aa(num,nat,numeral_numeral(nat),bit0(W2))) = $let(
            w: A,
            w:= aa(nat,A,power_power(A,Z2),aa(num,nat,numeral_numeral(nat),W2)),
            aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).

% power_numeral_even
tff(fact_1694_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R3),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_1695_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),one_one(A))),R3) ) ).

% of_int_ceiling_diff_one_le
tff(fact_1696_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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_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_1703_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_1704_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1705_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,aa(A,fun(A,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_1706_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),aa(A,A,aa(A,fun(A,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)),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_1707_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1708_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$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_1709_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,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$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_1710_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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$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_1711_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,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1712_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1713_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1714_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1715_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
         => ( aa(A,$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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1716_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),aa(A,A,aa(A,fun(A,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_1717_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)),aa(A,A,aa(A,fun(A,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),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_1718_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),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),Z2))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1719_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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),Na))),aa(nat,A,power_power(A,A2),Na)) ) ) ) ).

% power_Suc_less
tff(fact_1720_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2
tff(fact_1721_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2_right
tff(fact_1722_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_1723_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),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),Z2))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1724_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$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_1725_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$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_1726_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$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_1727_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$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_1728_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),aa(A,A,aa(A,fun(A,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_1729_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),aa(A,A,aa(A,fun(A,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_1730_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_1731_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)) = aa(A,A,aa(A,fun(A,A),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_1732_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1733_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z2))),B2) = $ite(Z2 = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1734_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1735_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z2: A,B2: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z2)),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1736_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z2: A,B2: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z2))),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1737_evenE,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),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4) ) ) ).

% evenE
tff(fact_1738_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))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B4 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1739_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))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1740_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))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1741_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_1742_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_1743_log__mult,axiom,
    ! [A2: real,X: 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)),X)
         => ( 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),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1744_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T2: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,T2))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_1745_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archimedean_ceiling(A,X) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A2)),one_one(A))),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_1746_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))
           => ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).

% ceiling_unique
tff(fact_1747_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).

% ceiling_correct
tff(fact_1748_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_1749_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X) ) ) ).

% le_ceiling_iff
tff(fact_1750_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2)
         => ( aa(A,$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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1751_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),aa(A,A,aa(A,fun(A,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_1752_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)),aa(A,A,aa(A,fun(A,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),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_1753_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),aa(A,A,aa(A,fun(A,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_1754_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),aa(A,A,aa(A,fun(A,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_1755_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$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_1756_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$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_1757_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$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_1758_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$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_1759_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R3: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R3),S)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(A,A,minus_minus(A,V2),U))),S))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_1760_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_1761_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))),aa(A,A,aa(A,fun(A,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),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_1762_even__two__times__div__two,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),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_1763_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,M: nat] :
          aa(nat,A,power_power(A,P3),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,power_power(A,P3),aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1764_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),A2) = aa(nat,A,power_power(A,A2),Na) ) ) ) ).

% power_minus_mult
tff(fact_1765_four__x__squared,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_1766_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M4)),X)),C2) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1767_linear__plus__1__le__power,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)),one_one(real))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_1768_Bernoulli__inequality,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality
tff(fact_1769_log__nat__power,axiom,
    ! [X: real,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,log(B2),aa(nat,real,power_power(real,X),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),X)) ) ) ).

% log_nat_power
tff(fact_1770_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(nat,real,power_power(real,X),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_realpow
tff(fact_1771_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: 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)),X)
             => ( aa(real,real,log(A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1772_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_1773_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))),aa(A,A,aa(A,fun(A,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),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_1774_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_1775_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X),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))),X)),Y)) ) ).

% power2_sum
tff(fact_1776_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D3))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,D3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1777_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Na)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_1778_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : aa(A,$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))),X)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),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_1779_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),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))),X)),Y)) ) ).

% power2_diff
tff(fact_1780_Bernoulli__inequality__even,axiom,
    ! [Na: nat,X: real] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_1781_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( aa(A,$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),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1782_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1783_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_1784_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_ge
tff(fact_1785_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_gt
tff(fact_1786_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_1787_VEBT__internal_Olog__ceil__idem,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),X)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,X)))) ) ) ).

% VEBT_internal.log_ceil_idem
tff(fact_1788_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_1789_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1790_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1791_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),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,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_1792_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_1793_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1794_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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),Na) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Na)),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),Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_1795_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_1796_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_1797_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_1798_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1799_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_1800_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_1801_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_numeral
tff(fact_1802_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_1803_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_1804_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_1805_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_1806_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_1807_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_1808_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_1809_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_1810_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_1811_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( dvd_dvd(A,aa(A,A,abs_abs(A),M),K)
        <=> dvd_dvd(A,M,K) ) ) ).

% abs_dvd_iff
tff(fact_1812_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( dvd_dvd(A,M,aa(A,A,abs_abs(A),K))
        <=> dvd_dvd(A,M,K) ) ) ).

% dvd_abs_iff
tff(fact_1813_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% abs_of_nat
tff(fact_1814_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1815_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1816_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W2: A,Z2: A] :
          ( ( powr(A,W2,Z2) = zero_zero(A) )
        <=> ( W2 = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_1817_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z2: A] : powr(A,zero_zero(A),Z2) = zero_zero(A) ) ).

% powr_0
tff(fact_1818_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),X)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),X)) ) ).

% of_int_abs
tff(fact_1819_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N2: int] : X = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_floor_cancel
tff(fact_1820_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_1821_semiring__norm_I13_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(Na)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ).

% semiring_norm(13)
tff(fact_1822_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_1823_semiring__norm_I12_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),Na) = Na ).

% semiring_norm(12)
tff(fact_1824_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,tanh(real),X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_1825_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% tanh_real_less_iff
tff(fact_1826_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% tanh_real_le_iff
tff(fact_1827_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_1828_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_1829_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_1830_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_1831_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_neg_numeral
tff(fact_1832_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_1833_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),Na)) = aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),Na)) ) ).

% abs_power_minus
tff(fact_1834_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1835_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_1836_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% mult_less_cancel2
tff(fact_1837_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          powr(A,X,zero_zero(A)) = $ite(X = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_1838_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_1839_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( ( K = zero_zero(nat) )
        | dvd_dvd(nat,M,Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1840_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1841_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% floor_numeral
tff(fact_1842_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,X,A2))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_1843_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,X)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_1844_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,X,A2)),powr(real,X,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_1845_num__double,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Na) = bit0(Na) ).

% num_double
tff(fact_1846_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% floor_of_nat
tff(fact_1847_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num] : aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),Na)) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% power_mult_numeral
tff(fact_1848_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_1849_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X) ) ).

% tanh_real_pos_iff
tff(fact_1850_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% tanh_real_nonneg_iff
tff(fact_1851_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_1852_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$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_1853_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$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_1854_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_1855_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1856_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% mult_le_cancel2
tff(fact_1857_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1858_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1859_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_1860_powr__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_1861_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( powr(real,X,one_one(real)) = X )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% powr_one_gt_zero_iff
tff(fact_1862_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,A2)),powr(real,X,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_1863_artanh__minus__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_1864_numeral__powr__numeral__real,axiom,
    ! [M: num,Na: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),Na)) ).

% numeral_powr_numeral_real
tff(fact_1865_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Na))
        <=> ( ( A2 != zero_zero(A) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1866_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_1867_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_1868_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ) ).

% zero_le_floor
tff(fact_1869_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_1870_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),X) ) ) ).

% numeral_le_floor
tff(fact_1871_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% zero_less_floor
tff(fact_1872_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_1873_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% floor_less_numeral
tff(fact_1874_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% one_le_floor
tff(fact_1875_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_less_one
tff(fact_1876_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_neg_numeral
tff(fact_1877_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_diff_numeral
tff(fact_1878_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,Na: nat] : archim6421214686448440834_floor(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ).

% floor_numeral_power
tff(fact_1879_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_1880_powr__log__cancel,axiom,
    ! [A2: real,X: 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)),X)
         => ( powr(real,A2,aa(real,real,log(A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_1881_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_1882_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = 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))),Na)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_1883_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: num,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W2)) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2)) ) ) ) ).

% power_even_abs_numeral
tff(fact_1884_powr__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( powr(real,X,aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),Na)) ) ) ).

% powr_numeral
tff(fact_1885_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = one_one(A) ) ).

% power_minus1_even
tff(fact_1886_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$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))),X) ) ) ).

% numeral_less_floor
tff(fact_1887_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_1888_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),X) ) ) ).

% one_less_floor
tff(fact_1889_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_1890_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X) ) ) ).

% neg_numeral_le_floor
tff(fact_1891_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_1892_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_1893_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_1894_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( 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),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,Na),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1895_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$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))),X) ) ) ).

% neg_numeral_less_floor
tff(fact_1896_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_1897_square__powr__half,axiom,
    ! [X: real] : powr(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,abs_abs(real),X) ).

% square_powr_half
tff(fact_1898_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_1899_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_1900_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_1901_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_1902_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_1903_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_1904_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_1905_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_1906_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_1907_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),Na)) = aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Na) ) ).

% power_abs
tff(fact_1908_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_1909_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1910_mult__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% mult_0
tff(fact_1911_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Na: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),M)),Na) ) ).

% power_mult
tff(fact_1912_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)) ).

% add_mult_distrib
tff(fact_1913_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% add_mult_distrib2
tff(fact_1914_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1915_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_1916_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_1917_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1918_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_1919_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_cube
tff(fact_1920_nat__mult__1,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Na) = Na ).

% nat_mult_1
tff(fact_1921_nat__mult__1__right,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),one_one(nat)) = Na ).

% nat_mult_1_right
tff(fact_1922_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_1923_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_1924_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Q3) ).

% div_mult2_eq
tff(fact_1925_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),Na)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)) ).

% diff_mult_distrib
tff(fact_1926_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% diff_mult_distrib2
tff(fact_1927_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_1928_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_1929_nat__times__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_times_as_int
tff(fact_1930_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,minus_minus(int,Z1),Z22)),W2) = aa(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_1931_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,minus_minus(int,Z1),Z22)) = aa(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_1932_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_1933_imult__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( Na = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_1934_nat__mult__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_1935_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_1936_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_1937_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_1938_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_1939_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X) ) ).

% of_int_floor_le
tff(fact_1940_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% floor_less_cancel
tff(fact_1941_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_1942_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ) ).

% abs_mult_less
tff(fact_1943_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,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,minus_minus(A,B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1944_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,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,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_1945_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,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,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_1946_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_1947_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_1948_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_1949_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_1950_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_1951_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_1952_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,X)),zero_zero(real)) ).

% powr_non_neg
tff(fact_1953_powr__less__mono2__neg,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,X,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_1954_powr__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_1955_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,X,Y)) ).

% powr_ge_pzero
tff(fact_1956_powr__less__cancel,axiom,
    ! [X: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,X,A2)),powr(real,X,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_1957_powr__less__mono,axiom,
    ! [A2: real,B2: real,X: 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)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,X,A2)),powr(real,X,B2)) ) ) ).

% powr_less_mono
tff(fact_1958_powr__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,A2)),powr(real,X,B2)) ) ) ).

% powr_mono
tff(fact_1959_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X)) ) ).

% floor_le_ceiling
tff(fact_1960_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z2))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1961_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_1962_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_1963_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1964_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_1965_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_1966_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
     => ( ( Na = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1967_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X)) ) ).

% floor_le_round
tff(fact_1968_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_1969_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_1970_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_1971_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1972_bezout__lemma__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat,X: nat,Y: nat] :
      ( dvd_dvd(nat,D3,A2)
     => ( dvd_dvd(nat,D3,B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D3) ) )
         => ? [X4: nat,Y3: nat] :
              ( dvd_dvd(nat,D3,A2)
              & dvd_dvd(nat,D3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = 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)),D3) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1973_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( dvd_dvd(nat,D5,A2)
      & dvd_dvd(nat,D5,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D5) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D5) ) ) ) ).

% bezout_add_nat
tff(fact_1974_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( dvd_dvd(nat,D5,A2)
      & dvd_dvd(nat,D5,B2)
      & ( ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D5 )
        | ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_1975_zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( Na = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Na = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_1976_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1977_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Na: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),Na))
     => ( ( K != zero_zero(int) )
       => dvd_dvd(int,M,Na) ) ) ).

% zdvd_mult_cancel
tff(fact_1978_zdvd__mono,axiom,
    ! [K: int,M: int,T2: int] :
      ( ( K != zero_zero(int) )
     => ( dvd_dvd(int,M,T2)
      <=> dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),T2)) ) ) ).

% zdvd_mono
tff(fact_1979_zdvd__period,axiom,
    ! [A2: int,D3: int,X: int,T2: int,C2: int] :
      ( dvd_dvd(int,A2,D3)
     => ( dvd_dvd(int,A2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2))
      <=> dvd_dvd(int,A2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D3))),T2)) ) ) ).

% zdvd_period
tff(fact_1980_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( dvd_dvd(int,K,aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)))
    <=> dvd_dvd(int,K,Na) ) ).

% zdvd_reduce
tff(fact_1981_tanh__real__lt__1,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),X)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_1982_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_1983_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E: A] :
              ( aa(A,$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),X)),E) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_1984_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_1985_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_1986_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_1987_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_1988_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_1989_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_1990_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Na)) ) ).

% zero_le_power_abs
tff(fact_1991_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_1992_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          aa(A,A,abs_abs(A),X3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),zero_zero(A)),aa(A,A,uminus_uminus(A),X3),X3) ) ).

% abs_if_raw
tff(fact_1993_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_1994_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),D3)))) ) ).

% abs_diff_triangle_ineq
tff(fact_1995_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,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_1996_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),R3)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3)) ) ) ) ).

% abs_diff_le_iff
tff(fact_1997_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),R3)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3)) ) ) ) ).

% abs_diff_less_iff
tff(fact_1998_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X) ) ) ).

% le_floor_iff
tff(fact_1999_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_2000_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% floor_less_iff
tff(fact_2001_powr__mono2_H,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,X,A2)) ) ) ) ).

% powr_mono2'
tff(fact_2002_powr__less__mono2,axiom,
    ! [A2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_2003_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% le_floor_add
tff(fact_2004_powr__inj,axiom,
    ! [A2: real,X: 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,X) = powr(real,A2,Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_2005_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( 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,X,Y)) ) ) ).

% gr_one_powr
tff(fact_2006_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_2007_powr__mono__both,axiom,
    ! [A2: real,B2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_2008_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,X,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_2009_powr__divide,axiom,
    ! [X: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_2010_powr__mult,axiom,
    ! [X: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_2011_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Na: nat] :
          ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,power_power(A,X),Na)) = aa(nat,int,power_power(int,archim6421214686448440834_floor(A,X)),Na) ) ) ) ).

% floor_power
tff(fact_2012_log__base__powr,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( A2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A2,B2)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),B2) ) ) ).

% log_base_powr
tff(fact_2013_log__powr,axiom,
    ! [X: real,B2: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B2),X)) ) ) ).

% log_powr
tff(fact_2014_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_powr
tff(fact_2015_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_2016_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),B2)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y2))),D5)
               => ( 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_2017_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V2: real] :
      ( ( X = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),U)),Y))),V2) ) ) ).

% sin_bound_lemma
tff(fact_2018_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_2019_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q3)),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_2020_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_2021_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> dvd_dvd(nat,M,Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2022_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => dvd_dvd(nat,M,Na) ) ) ).

% dvd_mult_cancel
tff(fact_2023_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_2024_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_2025_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_2026_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_2027_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2028_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_2029_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D5: nat,X4: nat,Y3: nat] :
          ( dvd_dvd(nat,D5,A2)
          & dvd_dvd(nat,D5,B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D5) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2030_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( Na = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2031_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P1,X4)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Z3)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P1,X4) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_2032_plusinfinity,axiom,
    ! [D3: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P2,X4)
          <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X4)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P2,X4) ) )
         => ( ? [X_1: int] : aa(int,$o,P2,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_2033_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_2034_tanh__real__gt__neg1,axiom,
    ! [X: 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),X)) ).

% tanh_real_gt_neg1
tff(fact_2035_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_2036_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R3)
         => aa(A,$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,R3)))),R3) ) ) ).

% of_nat_floor
tff(fact_2037_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,$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),X))) ) ).

% abs_add_one_gt_zero
tff(fact_2038_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),X)) = 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))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_2039_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,Na: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) ) ).

% floor_divide_of_nat_eq
tff(fact_2040_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_2041_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% of_int_leD
tff(fact_2042_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% of_int_lessD
tff(fact_2043_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,real,power_power(real,X),Na) ) ) ).

% powr_realpow
tff(fact_2044_powr__less__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),X)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),X)) ) ) ) ).

% powr_less_iff
tff(fact_2045_less__powr__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_2046_log__less__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),X)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_2047_less__log__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),X) ) ) ) ).

% less_log_iff
tff(fact_2048_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_2049_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_2050_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq
tff(fact_2051_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_2052_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_2053_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_gt_diff_one
tff(fact_2054_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_ge_diff_one
tff(fact_2055_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),B2)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y2))),D5)
               => ( 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_2056_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z2),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z2))))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z2),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_2057_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : 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))),Na)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_2058_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_2059_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_2060_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)))) ) ).

% dividend_less_times_div
tff(fact_2061_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Q3))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2062_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% mult_eq_if
tff(fact_2063_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_2064_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_2065_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2066_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2067_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2068_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M))
       => ( dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,Na),Q3))
        <=> dvd_dvd(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_2069_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: 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),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
         => ( 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),Q3),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2070_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: 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),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3))
     => ( 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),R3),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2071_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B6: int,Q4: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),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)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2072_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B6: int,Q4: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2073_q__pos__lemma,axiom,
    ! [B6: int,Q4: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_2074_incr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2075_decr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2076_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
           => ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).

% floor_unique
tff(fact_2077_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,X) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_2078_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T2: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,T2))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_2079_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archim6421214686448440834_floor(A,X))
        <=> aa(A,$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),Z2)),one_one(A))),X) ) ) ).

% less_floor_iff
tff(fact_2080_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_2081_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),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_2082_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_2083_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2084_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Na) = aa(nat,A,power_power(A,A2),Na) ) ) ) ).

% power_even_abs
tff(fact_2085_powr__neg__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X) ) ) ).

% powr_neg_one
tff(fact_2086_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,X,Y)) = powr(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_2087_le__log__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),X) ) ) ) ).

% le_log_iff
tff(fact_2088_log__le__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),X)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_2089_le__powr__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_2090_powr__le__iff,axiom,
    ! [B2: real,X: 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)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),X)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),X)) ) ) ) ).

% powr_le_iff
tff(fact_2091_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq2
tff(fact_2092_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_2093_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$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,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)),P3) ) ) ).

% floor_divide_lower
tff(fact_2094_split__zdiv,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_2095_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: 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),Q3)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2096_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: 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),Q3)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2097_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2098_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2099_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2100_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Na)),aa(nat,A,power_power(A,B2),Na)) ) ) ) ).

% power_mono_even
tff(fact_2101_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( 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),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_2102_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( 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),X),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),X)) ) ) ).

% ln_powr_bound2
tff(fact_2103_log__add__eq__powr,axiom,
    ! [B2: real,X: 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)),X)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2104_add__log__eq__powr,axiom,
    ! [B2: real,X: 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)),X)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2105_minus__log__eq__powr,axiom,
    ! [B2: real,X: 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)),X)
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2106_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$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))),Na))) ) ).

% zero_le_even_power'
tff(fact_2107_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$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,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_2108_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_2109_log__minus__eq__powr,axiom,
    ! [B2: real,X: 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)),X)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2110_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2111_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),Na))) ) ) ).

% powr_neg_numeral
tff(fact_2112_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_2113_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_2114_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2115_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),aa(int,A,ring_1_of_int(A),Na)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,X) = Na ) ) ) ).

% round_unique'
tff(fact_2116_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2117_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,power_power(real,X),aa(int,nat,nat2,I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,power_power(real,X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_2118_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2119_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% mult_less_iff1
tff(fact_2120_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Na)),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),Na),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),Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2121_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),X: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4)
             => aa(A,$o,aa(A,fun(A,$o),P,X4),aa(nat,A,power_power(A,X4),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),X)),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2122_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_ao(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_2123_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => summable(real,aTP_Lamp_ap(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_2124_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_2125_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_round(A,X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).

% round_altdef
tff(fact_2126_arctan__double,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,X)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2127_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_2128_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_2129_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,arctan,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_2130_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arctan,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X) ) ).

% zero_less_arctan_iff
tff(fact_2131_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_2132_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% zero_le_arctan_iff
tff(fact_2133_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_2134_zdvd1__eq,axiom,
    ! [X: int] :
      ( dvd_dvd(int,X,one_one(int))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2135_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z2)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_2136_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z2)),one_one(int))
    <=> ( Z2 = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2137_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( dvd_dvd(nat,Na,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Na),K) ) ).

% dvd_nat_abs_iff
tff(fact_2138_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( dvd_dvd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Na)
    <=> dvd_dvd(int,K,aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_abs_dvd_iff
tff(fact_2139_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_2140_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_2141_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% arctan_less_iff
tff(fact_2142_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone'
tff(fact_2143_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% arctan_le_iff
tff(fact_2144_abs__zmult__eq__1,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),Na)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2145_abs__div,axiom,
    ! [Y: int,X: int] :
      ( dvd_dvd(int,Y,X)
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),X)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_2146_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X)) ) ).

% frac_ge_0
tff(fact_2147_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,X)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2148_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2149_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z2))) = 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),Z2))) ).

% nat_abs_mult_distrib
tff(fact_2150_dvd__imp__le__int,axiom,
    ! [I: int,D3: int] :
      ( ( I != zero_zero(int) )
     => ( dvd_dvd(int,D3,I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D3)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_2151_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_2152_zdvd__mult__cancel1,axiom,
    ! [M: int,Na: int] :
      ( ( M != zero_zero(int) )
     => ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),Na),M)
      <=> ( aa(int,int,abs_abs(int),Na) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2153_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_2154_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L))
    <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_2155_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L)))
    <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_2156_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2157_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_2158_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),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,minus_minus(nat,B2),A2),aa(nat,nat,minus_minus(nat,A2),B2)) ).

% nat_abs_int_diff
tff(fact_2159_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => topological_monoseq(real,power_power(real,X)) ) ) ).

% monoseq_realpow
tff(fact_2160_incr__lemma,axiom,
    ! [D3: int,Z2: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z2))),one_one(int))),D3))) ) ).

% incr_lemma
tff(fact_2161_decr__lemma,axiom,
    ! [D3: int,X: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z2))),one_one(int))),D3))),Z2) ) ).

% decr_lemma
tff(fact_2162_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),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,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_2163_nat0__intermed__int__val,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2164_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_2165_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W2: num] : aa(real,A,real_Vector_of_real(A),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) ) ).

% of_real_neg_numeral
tff(fact_2166_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_aq(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_2167_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_ar(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_2168_of__real__of__nat__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Na: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% of_real_of_nat_eq
tff(fact_2169_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Na: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,power_power(real,X),Na)) = aa(nat,A,power_power(A,aa(real,A,real_Vector_of_real(A),X)),Na) ) ).

% of_real_power
tff(fact_2170_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W2: num] : aa(real,A,real_Vector_of_real(A),aa(num,real,numeral_numeral(real),W2)) = aa(num,A,numeral_numeral(A),W2) ) ).

% of_real_numeral
tff(fact_2171_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z2: 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)),Z2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_as(fun(nat,real),fun(real,fun(nat,real)),F2),Z2)) ) ) ) ) ).

% summable_power_series
tff(fact_2172_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_at(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_2173_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_au(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_2174_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_av(nat,A)) ) ).

% summable_zero
tff(fact_2175_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_2176_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( aa(real,A,real_Vector_of_real(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2177_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2178_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_ax(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2179_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_2180_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_2181_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_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_2182_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_2183_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_2184_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_2185_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_2186_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_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2187_summable__rabs__cancel,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bd(fun(nat,real),fun(nat,real),F2))
     => summable(real,F2) ) ).

% summable_rabs_cancel
tff(fact_2188_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_ar(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_2189_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_2190_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_be(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_2191_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,X: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2192_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_2193_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bg(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_2194_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),M: nat,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z2))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(A,fun(nat,A)),F2),Z2)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_2195_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N4: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),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_bd(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_2196_arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_ap(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_2197_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_2198_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M4: nat,N: nat] :
              ( aa(nat,$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,X5,N)),aa(nat,A,X5,M4)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_2199_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M4: nat,N: nat] :
              ( aa(nat,$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,X5,M4)),aa(nat,A,X5,N)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_2200_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = 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))),Na)))),comm_s3205402744901411588hammer(A,Z2,Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)) ) ).

% pochhammer_double
tff(fact_2201_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Na)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_2202_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bj(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2203_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_2204_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2205_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: nat,Na: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),X),Na) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,X,Na)) ) ).

% pochhammer_of_nat
tff(fact_2206_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X5)) = suminf(A,aTP_Lamp_be(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_2207_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_2208_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,Na)) ) ) ).

% pochhammer_pos
tff(fact_2209_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Na) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2210_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A2,Na) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2211_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_2212_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_2213_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_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_2214_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_2215_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F2)),C2) ) ) ) ).

% suminf_divide
tff(fact_2216_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_2217_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_2218_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_2219_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_2220_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_2221_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_2222_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bd(fun(nat,real),fun(nat,real),F2))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_bd(fun(nat,real),fun(nat,real),F2))) ) ).

% summable_rabs
tff(fact_2223_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$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,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_2224_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_2225_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2226_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Na: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2227_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Na) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Na)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2228_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,Na: nat,M: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,Na)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),Na)),M)) ) ).

% pochhammer_product'
tff(fact_2229_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2230_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_2231_binomial__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K5),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K5)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_2232_binomial__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K5)
     => ( 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))),K5)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K5)) ) ) ).

% binomial_mono
tff(fact_2233_binomial__maximum_H,axiom,
    ! [Na: 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))),Na)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Na)) ).

% binomial_maximum'
tff(fact_2234_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( comm_s3205402744901411588hammer(A,Z2,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_2235_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K5)
     => ( 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))),K5)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K5)) ) ) ).

% binomial_strict_mono
tff(fact_2236_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K5),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K5)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2237_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R3),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2238_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,minus_minus(A,B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2239_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,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_2240_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_2241_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Na),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% zero_less_binomial_iff
tff(fact_2242_choose__two,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_2243_binomial__n__0,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2244_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ).

% binomial_eq_0_iff
tff(fact_2245_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2246_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2247_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% binomial_le_pow2
tff(fact_2248_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
     => ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2249_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,binomial(Na),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_2250_binomial__le__pow,axiom,
    ! [R3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),R3)),aa(nat,nat,power_power(nat,Na),R3)) ) ).

% binomial_le_pow
tff(fact_2251_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Na),K)) ) ).

% zero_less_binomial
tff(fact_2252_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Na),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_2253_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2254_ln__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_bm(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_2255_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_bn(nat,real)) ).

% pi_series
tff(fact_2256_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] :
          comm_s3205402744901411588hammer(A,A2,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_bo(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_2257_VEBT__internal_Obit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : vEBT_VEBT_bit_concat(H,L,D3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D3))),L) ).

% VEBT_internal.bit_concat_def
tff(fact_2258_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),aa(int,int,aa(int,fun(int,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(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2259_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_2260_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_2261_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,modulo_modulo(A,A2,B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mod_trivial
tff(fact_2262_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_2263_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) ) ).

% Suc_mono
tff(fact_2264_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_eq
tff(fact_2265_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc_right
tff(fact_2266_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_2267_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_2268_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_2269_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_2270_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% Suc_le_mono
tff(fact_2271_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_2272_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_2273_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),Na)),K) ).

% Suc_diff_diff
tff(fact_2274_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_Suc_Suc
tff(fact_2275_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_2276_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)) ) ).

% mod_minus_minus
tff(fact_2277_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,Na)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_2278_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_2279_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_2280_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_2281_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_2282_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_2283_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_2284_zero__less__Suc,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Na)) ).

% zero_less_Suc
tff(fact_2285_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2286_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2287_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_2288_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_2289_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_2290_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_2291_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_2292_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_2293_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_2294_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_2295_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_2296_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc_right
tff(fact_2297_power__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_2298_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,power_power(nat,X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2299_dvd__1__left,axiom,
    ! [K: nat] : dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat)),K) ).

% dvd_1_left
tff(fact_2300_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( dvd_dvd(nat,M,aa(nat,nat,suc,zero_zero(nat)))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2301_diff__Suc__1,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),one_one(nat)) = Na ).

% diff_Suc_1
tff(fact_2302_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_2303_binomial__1,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),aa(nat,nat,suc,zero_zero(nat))) = Na ).

% binomial_1
tff(fact_2304_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_2305_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_2306_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_2307_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_2308_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_2309_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Na) ) ) ).

% one_le_mult_iff
tff(fact_2310_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2311_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2312_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2313_Suc__numeral,axiom,
    ! [Na: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Na)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% Suc_numeral
tff(fact_2314_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_2315_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_2316_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_2317_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_2318_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_2319_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc
tff(fact_2320_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc'
tff(fact_2321_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2322_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_2323_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% even_mod_2_iff
tff(fact_2324_even__Suc,axiom,
    ! [Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,Na))
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na) ) ).

% even_Suc
tff(fact_2325_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Na)))
    <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na) ) ).

% even_Suc_Suc_iff
tff(fact_2326_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_2327_zmod__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit0(V2)),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V2),aa(num,int,numeral_numeral(int),W2))) ).

% zmod_numeral_Bit0
tff(fact_2328_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_2329_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_2330_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2331_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2332_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_2333_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_2334_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ).

% one_less_nat_eq
tff(fact_2335_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_2336_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_2337_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( 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))),Na)) = 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))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2338_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,Na)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_mod
tff(fact_2339_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_2340_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_2341_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_eq
tff(fact_2342_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A5),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_2343_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_2344_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A2,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_2345_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_2346_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_2347_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_2348_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,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),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_2349_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_2350_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_2351_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_2352_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A5),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_2353_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_2354_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_2355_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% mod_minus_eq
tff(fact_2356_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A5: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A5,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A5),B2) ) ) ) ).

% mod_minus_cong
tff(fact_2357_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2)) ) ).

% mod_minus_right
tff(fact_2358_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,Na: nat] : modulo_modulo(A,aa(nat,A,power_power(A,modulo_modulo(A,A2,B2)),Na),B2) = modulo_modulo(A,aa(nat,A,power_power(A,A2),Na),B2) ) ).

% power_mod
tff(fact_2359_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_2360_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,Na: A] :
          ( dvd_dvd(A,K,M)
         => ( dvd_dvd(A,K,Na)
           => dvd_dvd(A,K,modulo_modulo(A,M,Na)) ) ) ) ).

% dvd_mod
tff(fact_2361_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_2362_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_2363_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_2364_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N: nat] : X != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_2365_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_2366_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_2367_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_2368_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2369_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_2370_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2371_nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$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,Na) ) ) ).

% nat_induct
tff(fact_2372_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X4),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X4: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X4),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ) ).

% diff_induct
tff(fact_2373_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_2374_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_2375_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_2376_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_2377_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_2378_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,P,X_1)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2379_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_2380_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_lessD
tff(fact_2381_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_2382_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( ( aa(nat,nat,suc,M) != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_2383_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_2384_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% less_SucI
tff(fact_2385_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_2386_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_2387_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_2388_All__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,I4) ) ) ) ).

% All_less_Suc
tff(fact_2389_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M)
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M5) ) ) ).

% Suc_less_eq2
tff(fact_2390_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_2391_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_SucD
tff(fact_2392_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_2393_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_2394_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_2395_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_2396_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) ).

% add_Suc_shift
tff(fact_2397_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc
tff(fact_2398_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_2399_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_leD
tff(fact_2400_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_SucE
tff(fact_2401_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_SucI
tff(fact_2402_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M6)
     => ? [M4: nat] : M6 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_2403_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        | ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_2404_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_2405_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_2406_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% full_nat_induct
tff(fact_2407_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,P,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_2408_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R2,Y3),Z)
               => aa(nat,$o,aa(nat,fun(nat,$o),R2,X4),Z) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R2,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2409_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_2410_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$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,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_2411_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_2412_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Na: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Na),I) = semiri8178284476397505188at_aux(A,Inc,Na,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_2413_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_2414_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_2415_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2416_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_2417_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2418_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D5: 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),D5)) ) ) ).

% mod_eqE
tff(fact_2419_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_add1_eq
tff(fact_2420_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_2421_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_2422_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_2423_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
        <=> dvd_dvd(A,C2,aa(A,A,minus_minus(A,A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_2424_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : dvd_dvd(A,B2,aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2))) ) ).

% dvd_minus_mod
tff(fact_2425_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,M,K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2426_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_2427_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_2428_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2429_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2430_zmod__eq__0D,axiom,
    ! [M: int,D3: int] :
      ( ( modulo_modulo(int,M,D3) = zero_zero(int) )
     => ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q5) ) ).

% zmod_eq_0D
tff(fact_2431_zmod__eq__0__iff,axiom,
    ! [M: int,D3: int] :
      ( ( modulo_modulo(int,M,D3) = zero_zero(int) )
    <=> ? [Q6: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q6) ) ).

% zmod_eq_0_iff
tff(fact_2432_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_2433_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_2434_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_2435_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_2436_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,M: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_2437_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_Suc
tff(fact_2438_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),Na)),A2) ) ).

% power_Suc2
tff(fact_2439_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less_eq(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_2440_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less_eq(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N5)),aa(nat,A,F2,Na)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_2441_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Na)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_2442_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2443_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ? [M3: nat] : Na = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_2444_All__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_2445_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% gr0_implies_Suc
tff(fact_2446_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2447_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_2448_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_2449_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ~ ! [Q5: nat] : Na != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ) ).

% less_natE
tff(fact_2450_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M))) ).

% less_add_Suc1
tff(fact_2451_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I))) ).

% less_add_Suc2
tff(fact_2452_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_2453_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_2454_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_2455_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na) ) ).

% Suc_leI
tff(fact_2456_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_eq
tff(fact_2457_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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_2458_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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,I) ) ) ) ).

% inc_induct
tff(fact_2459_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_lessD
tff(fact_2460_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_2461_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_Suc_eq_le
tff(fact_2462_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_2463_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_2464_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Na) ).

% Suc_eq_plus1_left
tff(fact_2465_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_2466_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_2467_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_2468_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na))) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_2469_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_2470_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc
tff(fact_2471_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_2472_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_2473_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_2474_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_2475_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N: nat] : Z2 != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2476_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z2: int] :
      ( ! [N: nat] : aa(int,$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,Z2) ) ) ).

% int_of_nat_induct
tff(fact_2477_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_2478_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_2479_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_2480_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_2481_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_2482_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q3: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2483_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_2484_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(8)
tff(fact_2485_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(6)
tff(fact_2486_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_2487_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_2488_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_2489_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_2490_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_2491_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_2492_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2493_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2494_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_2495_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_2496_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_2497_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_2498_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_2499_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc,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),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2500_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2501_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_2502_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_2503_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_2504_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_2505_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2506_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Na)) = aa(nat,A,power_power(A,B2),aa(nat,nat,suc,Na)) )
         => ( aa(A,$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_2507_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Na))),aa(nat,A,power_power(A,B2),aa(nat,nat,suc,Na)))
         => ( aa(A,$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_2508_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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,Na))) ) ) ).

% power_gt1
tff(fact_2509_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,Na: 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)),Na)
       => ( ( modulo_modulo(int,A3,Na) = zero_zero(int) )
         => ( ( modulo_modulo(int,B3,Na) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2510_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),modulo_modulo(int,A2,B2))) ).

% zmod_zminus1_eq_if
tff(fact_2511_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,modulo_modulo(int,A2,B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_2512_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_2513_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_2514_div__mod__decomp__int,axiom,
    ! [A3: int,Na: int] : A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Na)),Na)),modulo_modulo(int,A3,Na)) ).

% div_mod_decomp_int
tff(fact_2515_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_2516_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [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,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_2517_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% one_less_mult
tff(fact_2518_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_2519_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_2520_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_2521_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,Na),K)) ) ).

% power_gt_expt
tff(fact_2522_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_2523_realpow__pos__nth2,axiom,
    ! [A2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
          & ( aa(nat,real,power_power(real,R),aa(nat,nat,suc,Na)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_2524_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),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,Na),aa(A,A,aa(A,fun(A,A),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_2525_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),X: nat,M7: nat] :
      ( aa(nat,$o,P,X)
     => ( ! [X4: nat] :
            ( aa(nat,$o,P,X4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M7) )
       => ~ ! [M4: nat] :
              ( aa(nat,$o,P,M4)
             => ~ ! [X3: nat] :
                    ( aa(nat,$o,P,X3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M4) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2526_int__Suc,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ).

% int_Suc
tff(fact_2527_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_2528_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2)
    <=> ? [N2: nat] : Z2 = 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_2529_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Na)),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,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2530_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Na)),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,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2531_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Na),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,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_2532_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(A,fun(nat,A)),F2),Z2)) ) ) ).

% summable_powser_split_head
tff(fact_2533_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(A,fun(nat,A)),F2),Z2)) ) ) ).

% powser_split_head(3)
tff(fact_2534_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Na: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2535_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_2536_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_2537_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_2538_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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,Na))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_2539_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_2540_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,minus_minus(int,K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_2541_pi__half__neq__two,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != aa(num,real,numeral_numeral(real),bit0(one2)) ).

% pi_half_neq_two
tff(fact_2542_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_2543_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_2544_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_double_not_eq_double
tff(fact_2545_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% double_not_eq_Suc_double
tff(fact_2546_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_2547_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2548_real__of__int__div__aux,axiom,
    ! [X: int,D3: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),X)),aa(int,real,ring_1_of_int(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),D3))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),modulo_modulo(int,X,D3))),aa(int,real,ring_1_of_int(real),D3))) ).

% real_of_int_div_aux
tff(fact_2549_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = $ite(M = zero_zero(nat),Na,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% add_eq_if
tff(fact_2550_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na)) ) ).

% div_if
tff(fact_2551_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_2552_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q3)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2553_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,Na: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),Na))),comm_s3205402744901411588hammer(A,Z2,Na)) ) ).

% pochhammer_rec'
tff(fact_2554_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Na))) ) ).

% pochhammer_Suc
tff(fact_2555_Suc__nat__number__of__add,axiom,
    ! [V2: num,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),one2))),Na) ).

% Suc_nat_number_of_add
tff(fact_2556_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))) ).

% not_zle_0_negative
tff(fact_2557_negative__zless__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2558_negD,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),zero_zero(int))
     => ? [N: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2559_Suc__as__int,axiom,
    ! [X3: nat] : aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).

% Suc_as_int
tff(fact_2560_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq3
tff(fact_2561_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2562_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_2563_odd__iff__mod__2__eq__one,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))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2564_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F2)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_2565_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_eq_mod
tff(fact_2566_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: 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),Q3)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B2)
         => ( modulo_modulo(int,A2,B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2567_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: 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),Q3)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
         => ( modulo_modulo(int,A2,B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2568_split__zmod,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Na,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Na) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2569_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_2570_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,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_2571_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),modulo_modulo(int,aa(int,int,minus_minus(int,K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2572_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_2573_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,minus_minus(int,B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2574_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E2) ) ) ).

% nat_approx_posE
tff(fact_2575_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = modulo_modulo(int,K,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_def
tff(fact_2576_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( modulo_modulo(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_2577_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2578_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2579_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2580_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2581_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2582_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_2583_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q6)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,P,Q6) ) ) ) ).

% split_div'
tff(fact_2584_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_2585_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2586_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_2587_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),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_2588_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_2589_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_2590_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq4
tff(fact_2591_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),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),Na),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% div_exp_mod_exp_eq
tff(fact_2592_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_2593_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_2594_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,Na: 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)),Na)
       => 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),divide_divide(int),A3),Na)),
                $ite(modulo_modulo(int,B3,Na) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_2595_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),modulo_modulo(int,Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2596_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),modulo_modulo(int,Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2597_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_2598_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : 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))),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_2599_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_2600_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2601_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$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,Na) ) ) ) ).

% nat_bit_induct
tff(fact_2602_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_2603_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_2604_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_2605_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(nat,nat,binomial(Na),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_2606_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,binomial(Na),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2607_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_2608_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2609_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2610_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2611_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2)) = 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_bs(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))),Z2)) ) ) ) ).

% powser_split_head(1)
tff(fact_2612_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))),Z2) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_2613_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2614_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$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))),Na)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_2615_minus__pi__half__less__zero,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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_2616_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2617_arctan__bounded,axiom,
    ! [Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_2618_arctan__lbound,axiom,
    ! [Y: real] : 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_2619_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2620_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,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_2621_nat__ivt__aux,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2622_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,power_power(int,K),M)),K) = aa(nat,int,power_power(int,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_2623_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2624_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,Na))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_2625_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2626_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2627_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),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_2628_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z2: A,K6: real,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),K6)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H))),K6)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H)),Na)),aa(nat,A,power_power(A,Z2),Na))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z2),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,power_power(real,K6),aa(nat,nat,minus_minus(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2629_sin__cos__npi,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% sin_cos_npi
tff(fact_2630_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = $ite(Na = 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,minus_minus(nat,Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2631_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Na: 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,Na) = $ite(Na = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_2632_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))),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_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_2633_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2634_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( X = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_2635_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = X ) ).

% bit.double_compl
tff(fact_2636_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_2637_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( modulo_modulo(nat,M,Na) = M ) ) ).

% mod_less
tff(fact_2638_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2639_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_right
tff(fact_2640_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_left
tff(fact_2641_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( real_V7770717601297561774m_norm(A,X) = zero_zero(real) )
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2642_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2643_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_2644_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W2: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W2)) = aa(num,real,numeral_numeral(real),W2) ) ).

% norm_numeral
tff(fact_2645_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_2646_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2647_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2648_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2649_norm__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Na: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,real,semiring_1_of_nat(real),Na) ) ).

% norm_of_nat
tff(fact_2650_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_2651_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2652_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2653_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_2654_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_2655_norm__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W2: num] : real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = aa(num,real,numeral_numeral(real),W2) ) ).

% norm_neg_numeral
tff(fact_2656_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))),Na) = modulo_modulo(nat,aa(nat,nat,suc,M),Na) ).

% Suc_mod_mult_self1
tff(fact_2657_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))),Na) = modulo_modulo(nat,aa(nat,nat,suc,M),Na) ).

% Suc_mod_mult_self2
tff(fact_2658_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M)),Na) = modulo_modulo(nat,aa(nat,nat,suc,M),Na) ).

% Suc_mod_mult_self3
tff(fact_2659_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)),M)),Na) = modulo_modulo(nat,aa(nat,nat,suc,M),Na) ).

% Suc_mod_mult_self4
tff(fact_2660_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_2661_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_2662_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_2663_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_2664_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(real,A,real_Vector_of_real(A),pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_2665_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_2666_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_2667_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_2668_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,bit_ri4277139882892585799ns_not(A),A2))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% even_not_iff
tff(fact_2669_mod2__Suc__Suc,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_2670_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W2: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W2)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_2671_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W2)) ) ).

% norm_mult_numeral2
tff(fact_2672_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),W2))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W2)) ) ).

% norm_divide_numeral
tff(fact_2673_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),Na)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2674_sin__npi2,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Na))) = zero_zero(real) ).

% sin_npi2
tff(fact_2675_sin__npi,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_2676_sin__npi__int,axiom,
    ! [Na: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_npi_int
tff(fact_2677_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_2678_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% not_one_eq
tff(fact_2679_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2680_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2681_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2682_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2683_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_2684_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_2685_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_2686_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_2687_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2688_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_2689_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = cos(real,X) ).

% cos_periodic
tff(fact_2690_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = sin(real,X) ).

% sin_periodic
tff(fact_2691_cos__2pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = cos(real,X) ).

% cos_2pi_minus
tff(fact_2692_cos__npi2,axiom,
    ! [Na: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi2
tff(fact_2693_cos__npi,axiom,
    ! [Na: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi
tff(fact_2694_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_2695_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_2696_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,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_2697_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_2698_sin__2npi,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2699_cos__2npi,axiom,
    ! [Na: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2700_sin__2pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_2pi_minus
tff(fact_2701_sin__int__2pin,axiom,
    ! [Na: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_2702_cos__int__2pin,axiom,
    ! [Na: int] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),Na))) = one_one(real) ).

% cos_int_2pin
tff(fact_2703_cos__npi__int,axiom,
    ! [Na: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Na))) = $ite(dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Na),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ).

% cos_npi_int
tff(fact_2704_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_2705_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_2706_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_2707_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M,Na))),Na) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),Na) ).

% mod_Suc_Suc_eq
tff(fact_2708_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M,Na)),Na) = modulo_modulo(nat,aa(nat,nat,suc,M),Na) ).

% mod_Suc_eq
tff(fact_2709_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Na)),M) ).

% mod_less_eq_dividend
tff(fact_2710_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2711_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_2712_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_2713_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_2714_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_2715_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_2716_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_2717_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_2718_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,X))),cos(A,X)) ) ).

% sin_double
tff(fact_2719_sincos__principal__value,axiom,
    ! [X: 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,X) )
      & ( cos(real,Y3) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_2720_sin__x__le__x,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),X) ) ).

% sin_x_le_x
tff(fact_2721_sin__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),one_one(real)) ).

% sin_le_one
tff(fact_2722_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(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_2723_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,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_2724_cos__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,X)),one_one(real)) ).

% cos_le_one
tff(fact_2725_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      modulo_modulo(nat,aa(nat,nat,suc,M),Na) = $ite(aa(nat,nat,suc,modulo_modulo(nat,M,Na)) = Na,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,M,Na))) ).

% mod_Suc
tff(fact_2726_mod__induct,axiom,
    ! [P: fun(nat,$o),Na: nat,P3: nat,M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),P3)
         => ( ! [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,M) ) ) ) ) ).

% mod_induct
tff(fact_2727_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,M,Na)),Na) ) ).

% mod_less_divisor
tff(fact_2728_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [M4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M4),zero_zero(nat))
     => ( ! [M4: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M4,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M4),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_2729_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2730_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_2731_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_ge_zero
tff(fact_2732_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),aa(real,real,abs_abs(real),X)) ).

% abs_sin_x_le_abs_x
tff(fact_2733_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Na: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),Na)) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),Na) ) ).

% norm_power
tff(fact_2734_mod__eq__0D,axiom,
    ! [M: nat,D3: nat] :
      ( ( modulo_modulo(nat,M,D3) = zero_zero(nat) )
     => ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q5) ) ).

% mod_eq_0D
tff(fact_2735_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( modulo_modulo(nat,M,Na) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) ) ) ).

% mod_geq
tff(fact_2736_mod__if,axiom,
    ! [M: nat,Na: nat] :
      modulo_modulo(nat,M,Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),M,modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na)) ).

% mod_if
tff(fact_2737_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( modulo_modulo(nat,M,Na) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) ) ) ).

% le_mod_geq
tff(fact_2738_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,X))),real_V7770717601297561774m_norm(complex,X)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2739_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2)) ).

% complex_mod_triangle_ineq2
tff(fact_2740_cos__arctan__not__zero,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_2741_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A2,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_2742_sin__cos__le1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,X)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,X)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_2743_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_2744_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_2745_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bt(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_2746_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2747_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),bit_se2584673776208193580ke_bit(A,Na),bit_ri4674362597316999326ke_bit(A,M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_2748_sin__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero
tff(fact_2749_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),sin(real,X)) ) ).

% sin_x_ge_neg_x
tff(fact_2750_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_2751_sin__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_ge_zero
tff(fact_2752_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2753_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat,Z2: A] :
          ( ( aa(nat,A,power_power(A,W2),Na) = aa(nat,A,power_power(A,Z2),Na) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z2) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2754_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_2755_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_2756_sin__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,X)) ).

% sin_ge_minus_one
tff(fact_2757_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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)
           => ( ( cos(real,X) = cos(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_2758_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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_eq(real),cos(real,X)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_2759_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: 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_eq(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_2760_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R3: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),R3)
         => ( aa(real,$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),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R3),S)) ) ) ) ).

% norm_mult_less
tff(fact_2761_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_2762_cos__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,X)) ).

% cos_ge_minus_one
tff(fact_2763_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_2764_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Na)),Na) ) ).

% mod_le_divisor
tff(fact_2765_abs__sin__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),one_one(real)) ).

% abs_sin_le_one
tff(fact_2766_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E2) ) ) ).

% norm_triangle_lt
tff(fact_2767_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R3: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),R3)
         => ( aa(real,$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),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R3),S)) ) ) ) ).

% norm_add_less
tff(fact_2768_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R3: real,B2: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R3)
         => ( aa(real,$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),R3),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_2769_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : aa(real,$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),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2770_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$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),X),Y))),E2) ) ) ).

% norm_triangle_le
tff(fact_2771_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_2772_div__less__mono,axiom,
    ! [A3: nat,B3: nat,Na: 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)),Na)
       => ( ( modulo_modulo(nat,A3,Na) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B3,Na) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B3),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2773_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z2: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z2))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2774_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),Na))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),Na)) ) ).

% norm_power_ineq
tff(fact_2775_abs__cos__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,X))),one_one(real)) ).

% abs_cos_le_one
tff(fact_2776_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E2) ) ) ).

% norm_triangle_le_diff
tff(fact_2777_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z2: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z2))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2778_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))) ) ).

% norm_triangle_ineq4
tff(fact_2779_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y)))) ) ).

% norm_triangle_sub
tff(fact_2780_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,Na))
    <=> ~ dvd_dvd(nat,Na,M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2781_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,Na,Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ~ ! [S3: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_2782_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,Na,Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ~ ! [S3: nat] : Na != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_2783_nat__mod__eq__lemma,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,Na) = modulo_modulo(nat,Y,Na) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
       => ? [Q5: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2784_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,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_2785_norm__triangle__ineq2,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,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_2786_div__mod__decomp,axiom,
    ! [A3: nat,Na: nat] : A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Na)),Na)),modulo_modulo(nat,A3,Na)) ).

% div_mod_decomp
tff(fact_2787_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na),Q3))),modulo_modulo(nat,M,Na)) ).

% mod_mult2_eq
tff(fact_2788_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W2)),sin(A,Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W2),Z2))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_2789_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W2)),cos(A,Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2))),sin(A,aa(A,A,minus_minus(A,W2),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_2790_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W2)),sin(A,Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2))),sin(A,aa(A,A,minus_minus(A,W2),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_2791_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W2)),sin(A,Z2)) = 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,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_2792_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,minus_minus(A,sin(A,W2)),sin(A,Z2)) = 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,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_2793_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,minus_minus(A,cos(A,W2)),cos(A,Z2)) = 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,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Z2),W2)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_2794_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_2795_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : modulo_modulo(nat,M,Na) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na)) ).

% modulo_nat_def
tff(fact_2796_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,Na,Q3) )
      <=> dvd_dvd(nat,Q3,aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2797_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,X) = cos(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% sin_cos_eq
tff(fact_2798_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,X) = sin(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% cos_sin_eq
tff(fact_2799_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W2)) = aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,sin(A,W2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_2800_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N4: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
             => aa(real,$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_bu(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_2801_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),sin(A,X)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_2802_nat__mod__as__int,axiom,
    ! [X3: nat,Xa3: nat] : modulo_modulo(nat,X3,Xa3) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X3),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_mod_as_int
tff(fact_2803_not__int__div__2,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% not_int_div_2
tff(fact_2804_even__not__iff__int,axiom,
    ! [K: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K) ) ).

% even_not_iff_int
tff(fact_2805_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_2806_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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,X)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_2807_cos__monotone__0__pi,axiom,
    ! [Y: real,X: 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),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_2808_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,W2),Na) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2809_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),pi)
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_2810_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B2),D3)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2811_even__even__mod__4__iff,axiom,
    ! [Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
    <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2812_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),pi)
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_2813_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: 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_eq(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_2814_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),real_V7770717601297561774m_norm(A,X))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_2815_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,Na)))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2816_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,M,Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2817_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I4: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_2818_mod__nat__eqI,axiom,
    ! [R3: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R3),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),M)
       => ( dvd_dvd(nat,Na,aa(nat,nat,minus_minus(nat,M),R3))
         => ( modulo_modulo(nat,M,Na) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2819_norm__triangle__ineq3,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,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_2820_real__of__nat__div__aux,axiom,
    ! [X: nat,D3: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(nat,real,semiring_1_of_nat(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),D3))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,X,D3))),aa(nat,real,semiring_1_of_nat(real),D3))) ).

% real_of_nat_div_aux
tff(fact_2821_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),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) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),pi)
            & ( X = cos(real,T4) )
            & ( Y = sin(real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_2822_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_2823_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N4: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
             => aa(real,$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_2824_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,modulo_modulo(int,X,Y)) = modulo_modulo(nat,aa(int,nat,nat2,X),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_2825_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_2826_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2827_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_2828_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),X: A,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),real_V7770717601297561774m_norm(A,X))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),F2),Z2)) ) ) ) ).

% powser_inside
tff(fact_2829_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_2830_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real))
         => summable(A,power_power(A,X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_2831_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_2832_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_2833_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_2834_cos__is__zero,axiom,
    ? [X4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X4) = zero_zero(real) )
      & ! [Y2: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y2) = zero_zero(real) ) )
         => ( Y2 = X4 ) ) ) ).

% cos_is_zero
tff(fact_2835_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_2836_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bt(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_bt(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_2837_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: 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),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_2838_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),pi)
            & ( cos(real,X4) = Y )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),pi)
                  & ( cos(real,Y2) = Y ) )
               => ( Y2 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_2839_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2840_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K7) )
        <=> ? [N6: nat] :
            ! [N2: A] : aa(real,$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_2841_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K7) )
        <=> ? [N6: nat] :
            ! [N2: A] : aa(real,$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_2842_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_2843_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(real,A,real_Vector_of_real(A),B2)),aa(real,A,real_Vector_of_real(A),A2)))),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,B2),A2))) ) ).

% norm_of_real_diff
tff(fact_2844_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),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) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( X = cos(real,T4) )
              & ( Y = sin(real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_2845_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),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) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( X = cos(real,T4) )
          & ( Y = sin(real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_2846_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = modulo_modulo(nat,M,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_def
tff(fact_2847_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% even_signed_take_bit_iff
tff(fact_2848_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),real_V7770717601297561774m_norm(A,X))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,real)),F2),Z2)) ) ) ) ).

% powser_insidea
tff(fact_2849_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),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) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( X = cos(real,T4) )
               => ( Y != sin(real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_2850_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,X) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2851_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_2852_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,Na: 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)),Na)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Na)),
                $ite(modulo_modulo(nat,B3,Na) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B3),Na)) ) ) ).

% verit_le_mono_div
tff(fact_2853_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A,W2: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W2)),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Z2),M)),aa(nat,A,power_power(A,W2),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z2),W2)))) ) ) ) ).

% norm_power_diff
tff(fact_2854_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2855_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2856_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2857_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2858_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2859_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W2)),cos(A,Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W2),Z2))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_2860_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W2)),cos(A,Z2)) = 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,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W2),Z2)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_2861_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
         => ( summable(A,F2)
           => ? [N7: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),N8)))),R3) ) ) ) ) ).

% suminf_exist_split
tff(fact_2862_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,R0: real,A2: fun(nat,A),M7: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),R0)
           => ( ! [N: nat] : aa(real,$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))),M7)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bw(real,fun(fun(nat,A),fun(nat,real)),R3),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_2863_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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,X)) ) ) ).

% sin_gt_zero2
tff(fact_2864_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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,X)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_2865_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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))),X))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_2866_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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,X)) ) ) ).

% cos_gt_zero
tff(fact_2867_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,X) = sin(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_2868_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_2869_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),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),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_2870_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X2: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_2871_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_2872_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,cos(A,W2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_2873_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2874_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2875_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2876_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_2877_sin__le__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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,X)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_2878_sin__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,X)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_2879_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),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,X)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_2880_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),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),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),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,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_2881_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X4) = Y )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y2) = Y ) )
               => ( Y2 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_2882_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_2883_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,X)) ) ) ).

% cos_ge_zero
tff(fact_2884_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X2: nat] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X2: nat] : X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_2885_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_2886_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_2887_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2888_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_2889_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2890_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I4: int] :
          ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),I4)
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_2891_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ? [I4: int] :
          ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),I4)
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_2892_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),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,Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2893_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( ( sin(real,X) = zero_zero(real) )
       => ? [N: nat] :
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_2894_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2895_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( ( cos(real,X) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_2896_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2897_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),X))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_2898_triangle__def,axiom,
    ! [Na: nat] : nat_triangle(Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_2899_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_bx(A,fun(nat,A),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,minus_minus(A,one_one(A)),Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_2900_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_2901_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2902_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z2) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z2 != complex2(cos(real,T4),sin(real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_2903_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_2904_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_2905_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : aa(num,num,bit1,M) != bit0(Na) ).

% semiring_norm(89)
tff(fact_2906_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : bit0(M) != aa(num,num,bit1,Na) ).

% semiring_norm(88)
tff(fact_2907_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_2908_semiring__norm_I84_J,axiom,
    ! [Na: num] : one2 != aa(num,num,bit1,Na) ).

% semiring_norm(84)
tff(fact_2909_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_2910_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_2911_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(80)
tff(fact_2912_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(73)
tff(fact_2913_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_av(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_2914_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_2915_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),aa(num,num,bit1,Na)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(7)
tff(fact_2916_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),bit0(Na)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(9)
tff(fact_2917_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,Na))) ).

% semiring_norm(14)
tff(fact_2918_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),Na)) ).

% semiring_norm(15)
tff(fact_2919_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(72)
tff(fact_2920_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(81)
tff(fact_2921_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_2922_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),aa(num,num,bit1,Na)) ).

% semiring_norm(77)
tff(fact_2923_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit1
tff(fact_2924_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)),one2)) ).

% semiring_norm(10)
tff(fact_2925_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_2926_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_2927_semiring__norm_I4_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% semiring_norm(4)
tff(fact_2928_semiring__norm_I3_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(Na)) = aa(num,num,bit1,Na) ).

% semiring_norm(3)
tff(fact_2929_tan__npi,axiom,
    ! [Na: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_2930_tan__periodic__n,axiom,
    ! [X: real,Na: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),Na)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_2931_tan__periodic__nat,axiom,
    ! [X: real,Na: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_nat
tff(fact_2932_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)))) ).

% semiring_norm(16)
tff(fact_2933_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(74)
tff(fact_2934_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(79)
tff(fact_2935_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).

% xor_numerals(8)
tff(fact_2936_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_2937_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).

% xor_numerals(2)
tff(fact_2938_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_2939_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),X: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bg(fun(nat,A),fun(nat,A),A2)),X)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_2940_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_2941_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2942_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_2943_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na)))) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_2944_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V2: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),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),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2945_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic
tff(fact_2946_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_2947_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_2948_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_2949_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),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_2950_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_2951_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_2952_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2)),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V2),aa(num,int,numeral_numeral(int),W2)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2953_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_2954_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2955_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F2,N) = zero_zero(A)
         => aa(A,$o,sums(A,F2),zero_zero(A)) ) ) ).

% sums_0
tff(fact_2956_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S: A,T2: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( aa(A,$o,sums(A,F2),S)
           => ( aa(A,$o,sums(A,G),T2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S),T2) ) ) ) ) ).

% sums_le
tff(fact_2957_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),A2: real] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aTP_Lamp_be(fun(nat,real),fun(nat,A),X5)),aa(real,A,real_Vector_of_real(A),A2)) ) ) ).

% sums_of_real
tff(fact_2958_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_by(fun(nat,real),fun(nat,A),F2)),aa(real,A,real_Vector_of_real(A),C2))
        <=> aa(real,$o,sums(real,F2),C2) ) ) ).

% sums_of_real_iff
tff(fact_2959_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = zero_zero(complex) )
    <=> ( ( A2 = zero_zero(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_2960_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_2961_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2962_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2963_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_au(nat,fun(fun(nat,A),fun(nat,A)),I),F2)),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_2964_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_2965_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_2966_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ba(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_2967_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% sums_diff
tff(fact_2968_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ).

% sums_divide
tff(fact_2969_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% sums_minus
tff(fact_2970_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_2971_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = one_one(complex) )
    <=> ( ( A2 = one_one(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_2972_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_2973_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W2: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W2) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W2) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_2974_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bz(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3))
          <=> aa(A,$o,sums(A,F2),D3) ) ) ) ).

% sums_mult_iff
tff(fact_2975_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ca(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C2))
          <=> aa(A,$o,sums(A,F2),D3) ) ) ) ).

% sums_mult2_iff
tff(fact_2976_complex__of__real__def,axiom,
    ! [R3: real] : aa(real,complex,real_Vector_of_real(complex),R3) = complex2(R3,zero_zero(real)) ).

% complex_of_real_def
tff(fact_2977_complex__of__real__code,axiom,
    ! [X3: real] : aa(real,complex,real_Vector_of_real(complex),X3) = complex2(X3,zero_zero(real)) ).

% complex_of_real_code
tff(fact_2978_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa: real] :
      ( ( complex2(X,Y) = aa(real,complex,real_Vector_of_real(complex),Xa) )
    <=> ( ( X = Xa )
        & ( Y = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_2979_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ar(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),A2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_2980_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F2)),S)
           => aa(A,$o,sums(A,F2),S) ) ) ) ).

% sums_Suc_imp
tff(fact_2981_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F2)),S)
        <=> aa(A,$o,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_2982_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2)),L)
         => aa(A,$o,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_2983_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(nat,fun(fun(nat,A),fun(nat,A)),Na),F2)),S)
          <=> aa(A,$o,sums(A,F2),S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2984_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2985_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Na))) ).

% eval_nat_numeral(3)
tff(fact_2986_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_2987_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W2: num] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W2)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W2)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_2988_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2989_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2990_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(10)
tff(fact_2991_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2992_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_2993_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] :
          aa(nat,A,power_power(A,Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = $let(
            w: A,
            w:= aa(nat,A,power_power(A,Z2),aa(num,nat,numeral_numeral(nat),W2)),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),w)),w) ) ) ).

% power_numeral_odd
tff(fact_2994_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z2: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cd(nat,fun(A,fun(nat,A)),M),Z2)),aa(nat,A,power_power(A,Z2),M)) ) ).

% powser_sums_if
tff(fact_2995_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_bg(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2996_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Na) ) ).

% numeral_Bit1_div_2
tff(fact_2997_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: num] : ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))) ) ).

% odd_numeral
tff(fact_2998_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q3: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2999_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_3000_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_3001_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Na) ).

% Suc3_eq_add_3
tff(fact_3002_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))) ) ).

% not_numeral_Bit0_eq
tff(fact_3003_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_3004_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Na: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na)),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_3005_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_3006_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),Na) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_3007_tan__45,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_3008_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_3009_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => aa(A,$o,sums(A,power_power(A,C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3010_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_ce(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3011_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),aa(real,real,aa(real,fun(real,real),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),X4)) ) ) ).

% lemma_tan_total
tff(fact_3012_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),X)) ) ) ).

% tan_gt_zero
tff(fact_3013_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_3014_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),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),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3015_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( 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),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),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3016_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( 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),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),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),X)) ) ) ) ).

% tan_monotone
tff(fact_3017_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X4) = Y )
      & ! [Y2: real] :
          ( ( 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),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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y2) = Y ) )
         => ( Y2 = X4 ) ) ) ).

% tan_total
tff(fact_3018_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3019_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ).

% tan_inverse
tff(fact_3020_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3021_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_3022_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_3023_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( aa(real,$o,sums(real,G),X)
     => aa(real,$o,sums(real,aTP_Lamp_cf(fun(nat,real),fun(nat,real),G)),X) ) ).

% sums_if'
tff(fact_3024_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F2: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),X)
     => ( aa(real,$o,sums(real,F2),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cg(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) ) ) ).

% sums_if
tff(fact_3025_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),X)) ) ) ).

% tan_pos_pi2_le
tff(fact_3026_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_3027_tan__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),X)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3028_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),aa(real,real,aa(real,fun(real,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),aa(real,real,aa(real,fun(real,real),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),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3029_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),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),X)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3030_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),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),X))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3031_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),X) = Y )
         => ( aa(real,real,arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3032_arctan__tan,axiom,
    ! [X: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3033_arctan,axiom,
    ! [Y: real] :
      ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3034_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_3035_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),cos(A,X))) ) ).

% cos_treble_cos
tff(fact_3036_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3037_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3038_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3039_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ? [Z: real] :
          ( 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),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z) = X ) ) ) ).

% tan_total_pi4
tff(fact_3040_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_3041_machin__Euler,axiom,
    aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_3042_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_3043_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3044_time__vebt__buildup,axiom,
    ! [U: nat] :
      ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(n)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U)) ) ).

% time_vebt_buildup
tff(fact_3045_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_3046_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_3047_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_3048_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_3049_VEBT__internal_Obuildup__build__time,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_V8346862874174094_d_u_p(Na)),vEBT_V8646137997579335489_i_l_d(Na)) ).

% VEBT_internal.buildup_build_time
tff(fact_3050_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    vEBT_V8346862874174094_d_u_p(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_3051_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    vEBT_V8646137997579335489_i_l_d(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_3052_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_3053_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_3054_vebt__inst_Otime__vebt__buildup,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U)) ) ).

% vebt_inst.time_vebt_buildup
tff(fact_3055_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_3056_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_3057_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_3058_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ch(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(A,fun(nat,A)),C2),X)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ch(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_3059_sin__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_cj(real,fun(nat,real),X)),sin(real,X)) ).

% sin_paired
tff(fact_3060_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,Na)) = semiring_char_0_fact(A,Na) ) ).

% of_nat_fact
tff(fact_3061_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_3062_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3063_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_3064_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa(nat,nat,suc,Na) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Na = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_3065_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Na) )
    <=> ( pred_numeral(K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_3066_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_3067_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_3068_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Na))),semiring_char_0_fact(A,Na)) ) ).

% fact_Suc
tff(fact_3069_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_3070_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Na) ) ).

% less_numeral_Suc
tff(fact_3071_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_3072_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_3073_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Na) ) ).

% le_numeral_Suc
tff(fact_3074_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),Na) ).

% diff_numeral_Suc
tff(fact_3075_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,Na),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_3076_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),inc(Na)) ) ).

% minus_not_numeral_eq
tff(fact_3077_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_3078_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% fact_2
tff(fact_3079_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% add_neg_numeral_special(5)
tff(fact_3080_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3081_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% diff_numeral_special(5)
tff(fact_3082_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_3083_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_3084_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_3085_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_3086_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Na: nat] : semiring_char_0_fact(A,Na) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3087_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_ck(fun(nat,real),fun(nat,A),F2)),X3) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,F2),X3)) ) ).

% diffs_of_real
tff(fact_3088_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3089_num__induct,axiom,
    ! [P: fun(num,$o),X: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X4: num] :
            ( aa(num,$o,P,X4)
           => aa(num,$o,P,inc(X4)) )
       => aa(num,$o,P,X) ) ) ).

% num_induct
tff(fact_3090_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).

% add_inc
tff(fact_3091_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_zero
tff(fact_3092_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_gt_zero
tff(fact_3093_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Na)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3094_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_1
tff(fact_3095_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_cl(fun(nat,A),fun(nat,A),C2)),X3) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X3)) ) ).

% diffs_minus
tff(fact_3096_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ).

% fact_mono
tff(fact_3097_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => dvd_dvd(A,semiring_char_0_fact(A,Na),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_3098_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_3099_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_3100_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_3101_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_3102_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_3103_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ) ).

% fact_less_mono
tff(fact_3104_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: 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,Na)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))) ) ).

% fact_fact_dvd_fact
tff(fact_3105_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Na),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3106_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,Na),Na))) ) ).

% fact_le_power
tff(fact_3107_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_3108_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_3109_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_3110_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% minus_numeral_inc_eq
tff(fact_3111_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))),semiring_char_0_fact(A,Na)) ) ) ).

% choose_dvd
tff(fact_3112_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,C2),X3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X3))),aa(nat,A,C2,aa(nat,nat,suc,X3))) ) ).

% diffs_def
tff(fact_3113_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),X4))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_3114_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Na)),semiring_char_0_fact(real,Na))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% square_fact_le_2_fact
tff(fact_3115_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_3116_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_3117_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),Na,one_one(nat))) ) ).

% fact_code
tff(fact_3118_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3119_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3120_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),Na) = 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))),Na)),semiring_char_0_fact(A,Na)) ) ).

% pochhammer_same
tff(fact_3121_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))) ) ) ) ).

% fact_binomial
tff(fact_3122_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Na)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K)))) ) ) ) ).

% binomial_fact
tff(fact_3123_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_3124_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K6: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),K6)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),K6)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),X4)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3125_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_3126_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3127_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = 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(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))),Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Na))),semiring_char_0_fact(A,Na)) ) ).

% fact_double
tff(fact_3128_cos__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_cp(real,fun(nat,real),X)),cos(real,X)) ).

% cos_paired
tff(fact_3129_sin__coeff__def,axiom,
    ! [X3: nat] :
      sin_coeff(X3) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),X3),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,X3),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X3))) ).

% sin_coeff_def
tff(fact_3130_cos__coeff__def,axiom,
    ! [X3: nat] :
      cos_coeff(X3) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X3)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3131_sin__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3132_cos__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3133_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_3134_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,X) = aa(real,real,sqrt,Y) )
    <=> ( X = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3135_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3136_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3137_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% real_sqrt_less_iff
tff(fact_3138_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% real_sqrt_le_iff
tff(fact_3139_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = one_one(real) )
    <=> ( X = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3140_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3141_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_3142_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3143_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3144_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3145_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_3146_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3147_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_3148_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3149_real__sqrt__abs2,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),X)) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs2
tff(fact_3150_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A2)),aa(real,real,sqrt,A2)) = aa(real,real,abs_abs(real),A2) ).

% real_sqrt_mult_self
tff(fact_3151_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3152_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3153_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_3154_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_3155_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_3156_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(num,real,numeral_numeral(real),bit0(one2)) ).

% real_sqrt_four
tff(fact_3157_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_3158_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_3159_real__sqrt__abs,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs
tff(fact_3160_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% real_sqrt_pow2_iff
tff(fact_3161_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X ) ) ).

% real_sqrt_pow2
tff(fact_3162_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : aa(nat,real,power_power(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3163_fact__ge__self,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),semiring_char_0_fact(nat,Na)) ).

% fact_ge_self
tff(fact_3164_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ).

% fact_mono_nat
tff(fact_3165_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3166_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3167_real__sqrt__minus,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,X)) ).

% real_sqrt_minus
tff(fact_3168_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,sqrt,X)),K) ).

% real_sqrt_power
tff(fact_3169_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_3170_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3171_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_3172_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_gt_zero
tff(fact_3173_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_3174_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( ( aa(real,real,sqrt,X) = zero_zero(real) )
       => ( X = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3175_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_3176_sin__coeff__Suc,axiom,
    ! [Na: nat] : sin_coeff(aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% sin_coeff_Suc
tff(fact_3177_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_3178_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,X)) = aa(real,real,sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_3179_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3180_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => dvd_dvd(nat,M,semiring_char_0_fact(nat,Na)) ) ) ).

% dvd_fact
tff(fact_3181_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_3182_cos__coeff__Suc,axiom,
    ! [Na: nat] : cos_coeff(aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% cos_coeff_Suc
tff(fact_3183_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_3184_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_3185_fact__div__fact__le__pow,axiom,
    ! [R3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),R3)))),aa(nat,nat,power_power(nat,Na),R3)) ) ).

% fact_div_fact_le_pow
tff(fact_3186_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),K)))),aa(nat,nat,binomial(Na),K)) = semiring_char_0_fact(nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_3187_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_3188_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3189_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3190_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_3191_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3192_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3193_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),X),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_eq(real),aa(real,real,sqrt,X)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3194_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),aa(real,real,aa(real,fun(real,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_3195_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Y )
     => ( X = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3196_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = X )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3197_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,D3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3198_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3199_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3200_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_3201_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_45
tff(fact_3202_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_45
tff(fact_3203_tan__60,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3204_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),X),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,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3205_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3206_sqrt__even__pow2,axiom,
    ! [Na: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(real,real,sqrt,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3207_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_3208_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_3209_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3210_ln__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_3211_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3212_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_30
tff(fact_3213_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_60
tff(fact_3214_complex__norm,axiom,
    ! [X: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_norm
tff(fact_3215_arsinh__real__aux,axiom,
    ! [X: 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),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3216_real__sqrt__power__even,axiom,
    ! [Na: nat,X: real] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),Na) = aa(nat,real,power_power(real,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3217_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3218_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3219_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( powr(real,X,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_3220_tan__30,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% tan_30
tff(fact_3221_cos__x__y__le__one,axiom,
    ! [X: real,Y: 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),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),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)) ).

% cos_x_y_le_one
tff(fact_3222_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),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,X),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_3223_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3224_cos__arctan,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_3225_sin__arctan,axiom,
    ! [X: real] : sin(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_3226_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),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)),X)
         => ( 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,X),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_3227_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,X))
     => ( sin(real,X) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,cos(real,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3228_arctan__half,axiom,
    ! [X: real] : aa(real,real,arctan,X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3229_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa(nat,nat,binomial(Na),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(Na),aa(nat,nat,minus_minus(nat,Na),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Na),K)),one_one(nat)),Na,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_3230_cos__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( cos(real,aa(real,real,arcsin,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3231_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3232_sin__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( sin(real,aa(real,real,arccos,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3233_time__vebt__member,axiom,
    ! [T2: vEBT_VEBT,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),n) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% time_vebt_member
tff(fact_3234_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Na),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_concat_bit(Na,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),L))) ).

% concat_bit_Suc
tff(fact_3235_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3236_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_3237_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3238_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_concat_bit(Na,K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_3239_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_concat_bit(Na,K),L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_3240_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_concat_bit(Na,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) ).

% concat_bit_of_zero_2
tff(fact_3241_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3242_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3243_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_3244_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_3245_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arcsin_minus_1
tff(fact_3246_VEBT__internal_Ovalid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% VEBT_internal.valid_tree_deg_neq_0
tff(fact_3247_VEBT__internal_Ovalid__0__not,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% VEBT_internal.valid_0_not
tff(fact_3248_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,M: nat,L: int,R3: int] : aa(int,int,bit_concat_bit(Na,K),aa(int,int,bit_concat_bit(M,L),R3)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),aa(int,int,bit_concat_bit(Na,K),L)),R3) ).

% concat_bit_assoc
tff(fact_3249_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R3: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,bit_concat_bit(Na,R3),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),R3) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_3250_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,B2: int] : bit_concat_bit(Na,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),B2)) = bit_concat_bit(Na,B2) ).

% concat_bit_take_bit_eq
tff(fact_3251_VEBT__internal_Odeg__not__0,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% VEBT_internal.deg_not_0
tff(fact_3252_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_3253_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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_eq(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ).

% arccos_le_mono
tff(fact_3254_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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,real,arccos,X) = aa(real,real,arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3255_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3256_arcsin__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_3257_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3258_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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,real,arcsin,X) = aa(real,real,arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3259_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_3260_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_3261_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_3262_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_3263_arccos__cos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => ( aa(real,real,arccos,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3264_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3265_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),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,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3266_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_3267_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3268_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_3269_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3270_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3271_arccos__cos2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),X)
       => ( aa(real,real,arccos,cos(real,X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3272_arccos__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_3273_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3274_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3275_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_3276_arccos__le__pi2,axiom,
    ! [Y: 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_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3277_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),aa(real,real,aa(real,fun(real,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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3278_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3279_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_3280_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3281_arcsin__sin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3282_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3283_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),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_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3284_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,X)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3285_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,X))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3286_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3287_vebt__space__linear__bound,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,n)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n))) ) ).

% vebt_space_linear_bound
tff(fact_3288_time__vebt__insert,axiom,
    ! [T2: vEBT_VEBT,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),n) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% time_vebt_insert
tff(fact_3289_time__vebt__succ,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),n)))))) ) ).

% time_vebt_succ
tff(fact_3290_time__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),n)))))) ) ).

% time_vebt_pred
tff(fact_3291_time__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),n)))))) ) ).

% time_vebt_delete
tff(fact_3292_vebt__inst_Ovebt__space__linear__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% vebt_inst.vebt_space_linear_bound
tff(fact_3293_VEBT__internal_Ospace__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% VEBT_internal.space_bound
tff(fact_3294_vebt__inst_Otime__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% vebt_inst.time_vebt_delete
tff(fact_3295_vebt__inst_Otime__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% vebt_inst.time_vebt_pred
tff(fact_3296_vebt__inst_Otime__vebt__succ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% vebt_inst.time_vebt_succ
tff(fact_3297_member__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_3298_succ__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_3299_insert__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_3300_pred__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_3301_VEBT__internal_Odelete__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% VEBT_internal.delete_bound_size_univ
tff(fact_3302_invar__vebt__insert,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n))
       => vEBT_invar_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X),n) ) ) ).

% invar_vebt_insert
tff(fact_3303_VEBT__internal_Ospace__2__pow__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,one2))))),aa(real,real,minus_minus(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% VEBT_internal.space_2_pow_bound
tff(fact_3304_VEBT__internal_Ocnt__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))))))) ) ).

% VEBT_internal.cnt_bound
tff(fact_3305_succ__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_3306_pred__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_3307_VEBT__internal_Ocnt__non__neg,axiom,
    ! [T2: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)) ).

% VEBT_internal.cnt_non_neg
tff(fact_3308_VEBT__internal_Ospace__space_H,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2)) ).

% VEBT_internal.space_space'
tff(fact_3309_VEBT__internal_Ospace__cnt,axiom,
    ! [T2: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2))) ).

% VEBT_internal.space_cnt
tff(fact_3310_VEBT__internal_Ospace_H__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% VEBT_internal.space'_bound
tff(fact_3311_VEBT__internal_Ocnt__bound_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% VEBT_internal.cnt_bound'
tff(fact_3312_VEBT__internal_Ovalid__pres__insert,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => vEBT_invar_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X),Na) ) ) ).

% VEBT_internal.valid_pres_insert
tff(fact_3313_VEBT__internal_Odelete__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(T2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% VEBT_internal.delete_bound_size_univ'
tff(fact_3314_VEBT__internal_Ot__build__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% VEBT_internal.t_build_cnt
tff(fact_3315_VEBT__internal_Ot__buildup__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% VEBT_internal.t_buildup_cnt
tff(fact_3316_succ__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ) ).

% succ_bound_height
tff(fact_3317_invar__vebt__buildup,axiom,
    ( vEBT_invar_vebt(vEBT_vebt_buildup(n),n)
  <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),n) ) ).

% invar_vebt_buildup
tff(fact_3318_VEBT__internal_Odelete__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% VEBT_internal.delete_bound_height'
tff(fact_3319_VEBT__internal_Obuildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na) ) ).

% VEBT_internal.buildup_gives_valid
tff(fact_3320_vebt__inst_Oinvar__vebt__buildup,axiom,
    ! [Na: nat] :
      ( vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% vebt_inst.invar_vebt_buildup
tff(fact_3321_pred__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d2(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% pred_bound_height'
tff(fact_3322_succ_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% succ'_bound_height
tff(fact_3323_VEBT__internal_Otwo__powr__height__bound__deg,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => 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))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% VEBT_internal.two_powr_height_bound_deg
tff(fact_3324_VEBT__internal_Ocount__buildup,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na))) ).

% VEBT_internal.count_buildup
tff(fact_3325_VEBT__internal_Oheigt__uplog__rel,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ).

% VEBT_internal.heigt_uplog_rel
tff(fact_3326_VEBT__internal_Ocount__buildup_H,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) ).

% VEBT_internal.count_buildup'
tff(fact_3327_member__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_3328_VEBT__internal_Oheight__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T2: vEBT_VEBT] :
      ( ( U = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(T2,Deg)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% VEBT_internal.height_double_log_univ_size
tff(fact_3329_VEBT__internal_Odelete__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% VEBT_internal.delete_bound_height
tff(fact_3330_pred__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ).

% pred_bound_height
tff(fact_3331_insert__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))) ) ).

% insert_bound_height
tff(fact_3332_cot__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),X)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_3333_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_cq(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_3334_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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_3335_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq1
tff(fact_3336_set__decode__0,axiom,
    ! [X: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),nat_set_decode(X))
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),X) ) ).

% set_decode_0
tff(fact_3337_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_3338_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_3339_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_3340_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_3341_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_3342_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_3343_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_3344_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_3345_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_3346_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_3347_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% exp_less_cancel_iff
tff(fact_3348_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_3349_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% exp_le_cancel_iff
tff(fact_3350_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_3351_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3352_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_3353_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_3354_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,minus_minus(A,one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_3355_Suc__0__mod__eq,axiom,
    ! [Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa($o,nat,zero_neq_one_of_bool(nat),Na != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_3356_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_3357_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_3358_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_3359_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),X) = one_one(real) )
    <=> ( X = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_3360_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_3361_exp__less__one__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_3362_one__less__exp__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X) ) ).

% one_less_exp_iff
tff(fact_3363_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3364_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,exp(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% one_le_exp_iff
tff(fact_3365_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_3366_exp__ln,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_3367_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X) ) ).

% exp_ln_iff
tff(fact_3368_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P3: $o] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> (P3) ) ) ).

% odd_of_bool_self
tff(fact_3369_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_3370_cot__npi,axiom,
    ! [Na: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_3371_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($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_3372_set__decode__Suc,axiom,
    ! [Na: nat,X: nat] :
      ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Na)),nat_set_decode(X))
    <=> aa(set(nat),$o,member(nat,Na),nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_3373_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cot(real),X) ).

% cot_periodic
tff(fact_3374_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_3375_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_3376_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_of_exp
tff(fact_3377_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_3378_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : modulo_modulo(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_3379_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),X))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,X))) ) ).

% norm_exp
tff(fact_3380_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% exp_less_cancel
tff(fact_3381_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o,Q3: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = aa($o,A,zero_neq_one_of_bool(A),(Q3)) )
        <=> ( (P3)
          <=> (Q3) ) ) ) ).

% of_bool_eq_iff
tff(fact_3382_of__nat__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,Na),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) ) ).

% of_nat_gbinomial
tff(fact_3383_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3384_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_3385_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% subset_decode_imp_le
tff(fact_3386_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X4: real] : aa(real,real,exp(real),X4) = Y ) ).

% exp_total
tff(fact_3387_exp__gt__zero,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,exp(real),X)) ).

% exp_gt_zero
tff(fact_3388_not__exp__less__zero,axiom,
    ! [X: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),X)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_3389_exp__ge__zero,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,exp(real),X)) ).

% exp_ge_zero
tff(fact_3390_not__exp__le__zero,axiom,
    ! [X: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3391_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_3392_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_3393_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_3394_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_3395_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_3396_binomial__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) ) ).

% binomial_gbinomial
tff(fact_3397_exp__gt__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),X)) ) ).

% exp_gt_one
tff(fact_3398_exp__ge__add__one__self,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X)) ).

% exp_ge_add_one_self
tff(fact_3399_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,X: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),X)) = aa(nat,A,power_power(A,aa(A,A,exp(A),X)),Na) ) ).

% exp_of_nat_mult
tff(fact_3400_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Na: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(nat,A,power_power(A,aa(A,A,exp(A),X)),Na) ) ).

% exp_of_nat2_mult
tff(fact_3401_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3402_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3403_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( aa(real,real,exp(real),X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3404_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Y)),X) ) ) ).

% ln_ge_iff
tff(fact_3405_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),X)) ) ) ).

% ln_x_over_x_mono
tff(fact_3406_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_3407_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_3408_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3409_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3410_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A2: A] :
          powr(A,X,A2) = $ite(X = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),X)))) ) ).

% powr_def
tff(fact_3411_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_3412_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa($o,A,zero_neq_one_of_bool(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))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_3413_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_3414_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3415_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3416_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,semiring_1_of_nat(A),Na)))),Na) = aa(A,A,exp(A),X) ) ) ) ).

% exp_divide_power_eq
tff(fact_3417_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% exp_half_le2
tff(fact_3418_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,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)
               => ( ( aa(A,A,aa(A,fun(A,A),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_3419_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3420_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_3421_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : 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,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na))),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))),Na)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Na)) ) ).

% gbinomial_index_swap
tff(fact_3422_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3423_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z2)) = aa(nat,A,power_power(A,aa(A,A,exp(A),Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_3424_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : modulo_modulo(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_3425_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,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_3426_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,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3427_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3428_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z2))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_3429_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,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_3430_exp__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_3431_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_3432_xor__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = 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)),M) != ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na))),
        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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_3433_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($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_3434_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($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_3435_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($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_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_3436_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_3437_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),X)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3438_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na))) ).

% xor_Suc_0_eq
tff(fact_3439_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na))) ).

% Suc_0_xor_eq
tff(fact_3440_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))) ) ).

% exp_div_exp_eq
tff(fact_3441_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3442_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3443_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z2)))) ) ) ).

% exp_bound_lemma
tff(fact_3444_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),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),X)) ) ) ).

% cot_gt_zero
tff(fact_3445_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),X)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_3446_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_3447_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z2: nat] :
      ( ~ aa(set(nat),$o,member(nat,Na),nat_set_decode(Z2))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Z2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),nat_set_decode(Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_3448_tan__cot_H,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,cot(real),X) ).

% tan_cot'
tff(fact_3449_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))) ).

% tanh_real_altdef
tff(fact_3450_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_cr(nat,fun(nat,$o),X)) ).

% set_decode_def
tff(fact_3451_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A2: A,B2: A,C2: A,D3: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3452_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_cs(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_3453_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_ct(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_3454_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Na)) = 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(Na)))) ) ).

% mask_numeral
tff(fact_3455_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_3456_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_3457_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3458_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Na) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_3459_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_3460_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,Na) ) ).

% take_bit_minus_one_eq_mask
tff(fact_3461_nat__mask__eq,axiom,
    ! [Na: nat] : aa(int,nat,nat2,bit_se2239418461657761734s_mask(int,Na)) = bit_se2239418461657761734s_mask(nat,Na) ).

% nat_mask_eq
tff(fact_3462_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,Na)) = bit_se2239418461657761734s_mask(A,Na) ) ).

% of_nat_mask_eq
tff(fact_3463_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,Na)) = bit_se2239418461657761734s_mask(A,Na) ) ).

% of_int_mask_eq
tff(fact_3464_less__eq__mask,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ).

% less_eq_mask
tff(fact_3465_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Na)) ).

% mask_nonnegative_int
tff(fact_3466_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_3467_ln__real__def,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_cu(real,fun(real,$o),X)) ).

% ln_real_def
tff(fact_3468_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = the(A,sums(A,F2)) ) ).

% suminf_def
tff(fact_3469_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ) ).

% less_mask
tff(fact_3470_ln__neg__is__const,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_cv(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_3471_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,Na)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3472_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = bit_se2239418461657761734s_mask(int,Na) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_3473_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_3474_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3475_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,Na)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_mask_eq_exp
tff(fact_3476_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% mask_nat_less_exp
tff(fact_3477_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_cw(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_3478_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se2239418461657761734s_mask(A,Na))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_3479_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_3480_mask__nat__def,axiom,
    ! [Na: nat] : bit_se2239418461657761734s_mask(nat,Na) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)) ).

% mask_nat_def
tff(fact_3481_mask__half__int,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,Na)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ).

% mask_half_int
tff(fact_3482_mask__int__def,axiom,
    ! [Na: nat] : bit_se2239418461657761734s_mask(int,Na) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) ).

% mask_int_def
tff(fact_3483_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(A,Na) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_3484_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)) ) ).

% minus_exp_eq_not_mask
tff(fact_3485_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_cx(real,$o)) ).

% pi_half
tff(fact_3486_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),the(real,aTP_Lamp_cx(real,$o))) ).

% pi_def
tff(fact_3487_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = bit_se2239418461657761734s_mask(int,Na) )
    <=> dvd_dvd(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_3488_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_3489_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% the_sym_eq_trivial
tff(fact_3490_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_cy(A,fun(A,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_3491_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_3492_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo(int,K,L) = $ite(
        L = zero_zero(int),
        K,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_3493_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_3494_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ).

% sgn_sgn
tff(fact_3495_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3496_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_3497_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_3498_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3499_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,power_power(A,A2),Na)) = aa(nat,A,power_power(A,aa(A,A,sgn_sgn(A),A2)),Na) ) ).

% power_sgn
tff(fact_3500_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_3501_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_3502_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)
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3503_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Na) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_3504_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Na) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_3505_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3506_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_3507_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3508_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_3509_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),L),K)
    <=> ( dvd_dvd(int,L,K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3510_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R3)),K)
    <=> ( dvd_dvd(int,L,K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3511_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),K))
    <=> ( dvd_dvd(int,L,K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3512_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R3)))
    <=> ( dvd_dvd(int,L,K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3513_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3514_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_3515_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))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3516_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% sgn_of_nat
tff(fact_3517_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,Na: num] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W2))),aa(num,nat,numeral_numeral(nat),Na))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),pred_numeral(Na)) ) ) ).

% bit_numeral_simps(2)
tff(fact_3518_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_3519_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,Na: num] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W2))),aa(num,nat,numeral_numeral(nat),Na))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),pred_numeral(Na)) ) ) ).

% bit_numeral_simps(3)
tff(fact_3520_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(nat,nat,suc,Na))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W2)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_3521_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_3522_bit__minus__numeral__int_I1_J,axiom,
    ! [W2: num,Na: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(num,nat,numeral_numeral(nat),Na))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(1)
tff(fact_3523_bit__minus__numeral__int_I2_J,axiom,
    ! [W2: num,Na: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(num,nat,numeral_numeral(nat),Na))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W2)),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(2)
tff(fact_3524_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_3525_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
            & ( M != Na ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_3526_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$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)),Na)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Na) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_3527_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_3528_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),Na)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,M),Na) ) ) ).

% bit_of_nat_iff_bit
tff(fact_3529_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,sgn_sgn(A),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_3530_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Na)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),M)),Na) ) ) ).

% bit_numeral_iff
tff(fact_3531_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_3532_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3533_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3534_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),Na)
        <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
            <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Na) ) ) ) ).

% bit_xor_iff
tff(fact_3535_bit__not__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),Na)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% bit_not_int_iff
tff(fact_3536_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),Na)
    <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Na) ) ) ).

% bit_xor_int_iff
tff(fact_3537_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,Na)) ) ).

% not_bit_1_Suc
tff(fact_3538_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% bit_numeral_simps(1)
tff(fact_3539_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Na)
        <=> ( Na = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3540_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_3541_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3542_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B2))),Na)
        <=> ( (B2)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3543_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_3544_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3545_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,sgn_sgn(A),A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_3546_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_3547_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_3548_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_3549_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_sgnE
tff(fact_3550_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_3551_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_3552_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_3553_bit__not__int__iff_H,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int))),Na)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% bit_not_int_iff'
tff(fact_3554_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ dvd_dvd(int,L,K)
       => ( aa(int,int,sgn_sgn(int),modulo_modulo(int,K,L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_3555_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          bit_se8732182000553998342ip_bit(A,Na,A2) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Na),
            A2) ) ).

% flip_bit_eq_if
tff(fact_3556_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_3557_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          aa(A,A,sgn_sgn(A),X) = $ite(
            X = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_3558_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3559_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = $ite(X = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3560_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3561_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3562_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( dvd_dvd(int,L,K)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_3563_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_3564_bit__minus__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),Na)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,minus_minus(int,K),one_one(int)))),Na) ) ).

% bit_minus_int_iff
tff(fact_3565_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,bit_concat_bit(Na,K),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_3566_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3567_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ) ).

% bit_Suc
tff(fact_3568_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
          <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_3569_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_3570_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M2)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_3571_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_3572_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X3 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_3573_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A2 ) )
       => ( ! [X4: A] :
              ( aa(A,$o,P,X4)
             => aa(A,$o,Q,X4) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_3574_If__def,axiom,
    ! [A: $tType,P: $o,X: A,Y: A] :
      $ite((P),X,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_cz($o,fun(A,fun(A,fun(A,$o))),(P)),X),Y)) ).

% If_def
tff(fact_3575_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X3 ) ) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_3576_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X3 ) ) )
     => ( aa(A,$o,P,A2)
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_3577_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ).

% bit_iff_odd
tff(fact_3578_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Na)
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_3579_bit__int__def,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% bit_int_def
tff(fact_3580_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Na)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3581_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A2),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3582_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3583_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( ! [J2: nat] : ~ aa(nat,$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))),Na)
          <=> $ite(Na = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),aa(nat,$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)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3584_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
        <=> $ite(Na = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_3585_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_da(real,fun(int,$o),X)) ).

% floor_real_def
tff(fact_3586_set__bit__eq,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% set_bit_eq
tff(fact_3587_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% unset_bit_eq
tff(fact_3588_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ).

% take_bit_Suc_from_most
tff(fact_3589_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Na,M)))))) ) ).

% divide_int_unfold
tff(fact_3590_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Na))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Na,M))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Na))))) ) ).

% modulo_int_unfold
tff(fact_3591_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(int,L,K)))))) ) ).

% divide_int_def
tff(fact_3592_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% and_int_unfold
tff(fact_3593_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_3594_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_3595_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),X)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,X)) ) ) ).

% arctan_inverse
tff(fact_3596_member__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% member_bound_height'
tff(fact_3597_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_3598_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_3599_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_3600_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_3601_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_3602_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_3603_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_3604_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2)) ) ).

% take_bit_and
tff(fact_3605_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A2) = A2 ) ).

% and.left_neutral
tff(fact_3606_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = A2 ) ).

% and.right_neutral
tff(fact_3607_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_3608_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_3609_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_3610_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_3611_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_3612_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% zero_le_sgn_iff
tff(fact_3613_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sgn_sgn(real),X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_3614_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_3615_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_3616_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_3617_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_3618_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_3619_and__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_3620_and__minus__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_3621_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_3622_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_3623_and__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_3624_and__minus__numerals_I1_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_3625_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_3626_bit__and__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),Na)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Na) ) ) ).

% bit_and_int_iff
tff(fact_3627_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),Na)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Na) ) ) ) ).

% bit_and_iff
tff(fact_3628_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_and_eq
tff(fact_3629_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_3630_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_3631_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_3632_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_3633_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z2),X)) ) ).

% bit.conj_xor_distrib2
tff(fact_3634_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z2)) ) ).

% bit.conj_xor_distrib
tff(fact_3635_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_3636_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3637_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_3638_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)) ) ).

% AND_lower
tff(fact_3639_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X) ) ).

% AND_upper1
tff(fact_3640_AND__upper2,axiom,
    ! [Y: int,X: 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_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y) ) ).

% AND_upper2
tff(fact_3641_AND__upper1_H,axiom,
    ! [Y: int,Z2: 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_eq(int),Y),Z2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z2) ) ) ).

% AND_upper1'
tff(fact_3642_AND__upper2_H,axiom,
    ! [Y: int,Z2: int,X: 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_eq(int),Y),Z2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z2) ) ) ).

% AND_upper2'
tff(fact_3643_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,Na)) ) ).

% take_bit_eq_mask
tff(fact_3644_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Na)) ).

% not_bit_Suc_0_numeral
tff(fact_3645_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N)
             => aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N) )
         => ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_3646_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_3647_AND__upper1_H_H,axiom,
    ! [Y: int,Z2: 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),Z2)
       => 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)),Z2) ) ) ).

% AND_upper1''
tff(fact_3648_AND__upper2_H_H,axiom,
    ! [Y: int,Z2: int,X: 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),Z2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z2) ) ) ).

% AND_upper2''
tff(fact_3649_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_3650_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(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_and_iff
tff(fact_3651_sgn__real__def,axiom,
    ! [A2: real] :
      aa(real,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_3652_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
        | dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ) ) ).

% even_and_iff_int
tff(fact_3653_and__not__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_3654_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(4)
tff(fact_3655_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),Na)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ) ).

% bit_nat_iff
tff(fact_3656_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),one_one(A)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_3657_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_3658_sgn__power__injE,axiom,
    ! [A2: real,Na: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),A2)),Na)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),B2)),Na)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_3659_and__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = 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),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(5)
tff(fact_3660_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(7)
tff(fact_3661_and__not__numerals_I3_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_3662_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3663_bit__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,M),Na)
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% bit_nat_def
tff(fact_3664_and__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = 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),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(9)
tff(fact_3665_and__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = 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),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(6)
tff(fact_3666_and__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% and_not_numerals(8)
tff(fact_3667_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_3668_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_db(rat,fun(int,$o),X)) ).

% floor_rat_def
tff(fact_3669_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( aa(set(int),$o,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & aa(set(int),$o,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ 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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_3670_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( Y = $ite(
            ( aa(set(int),$o,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
                & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
            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)),X)
                  & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
              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),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3671_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3672_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).

% bot_apply
tff(fact_3673_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
    <=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).

% empty_Collect_eq
tff(fact_3674_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).

% Collect_empty_eq
tff(fact_3675_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X2: A] : ~ aa(set(A),$o,member(A,X2),A3)
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_3676_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ aa(set(A),$o,member(A,C2),bot_bot(set(A))) ).

% empty_iff
tff(fact_3677_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_3678_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_3679_singletonI,axiom,
    ! [A: $tType,A2: A] : aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_3680_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_3681_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_3682_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_3683_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3684_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cy(A,fun(A,$o),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3685_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$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)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3686_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3687_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_3688_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) ).

% insert_Diff_single
tff(fact_3689_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
    <=> ~ aa(set(A),$o,member(A,B2),A3) ) ).

% subset_Compl_singleton
tff(fact_3690_divide__numeral__i,axiom,
    ! [Z2: complex,Na: num] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z2),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),Na)),imaginary_unit)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z2))),aa(num,complex,numeral_numeral(complex),Na)) ).

% divide_numeral_i
tff(fact_3691_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_3692_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_3693_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_3694_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_3695_power2__i,axiom,
    aa(nat,complex,power_power(complex,imaginary_unit),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_3696_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_3697_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_3698_i__even__power,axiom,
    ! [Na: nat] : aa(nat,complex,power_power(complex,imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,complex,power_power(complex,aa(complex,complex,uminus_uminus(complex),one_one(complex))),Na) ).

% i_even_power
tff(fact_3699_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),bit0(one2))),aa(real,complex,real_Vector_of_real(complex),pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3700_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,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_3701_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),X),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),X),Y)
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_3702_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_3703_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_3704_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_3705_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X4: A] : ~ aa(set(A),$o,member(A,X4),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_3706_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_3707_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_3708_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D3 ) )
        | ( ( A2 = D3 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3709_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_3710_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_3711_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_3712_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_3713_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_dc(A,$o)) ).

% empty_def
tff(fact_3714_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_3715_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X2: A] : aa(set(A),$o,member(A,X2),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_3716_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y3: A] : ~ aa(set(A),$o,member(A,Y3),A3)
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_3717_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ aa(set(A),$o,member(A,A2),A3) ) ).

% equals0D
tff(fact_3718_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ aa(set(A),$o,member(A,A2),bot_bot(set(A))) ).

% emptyE
tff(fact_3719_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_3720_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
     => ~ ! [S3: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S3)
           => ! [T4: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T4)
               => ( R3 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3721_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_3722_complex__i__not__numeral,axiom,
    ! [W2: num] : imaginary_unit != aa(num,complex,numeral_numeral(complex),W2) ).

% complex_i_not_numeral
tff(fact_3723_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_dd(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_3724_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_de(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_3725_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,X),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% diff_shunt_var
tff(fact_3726_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$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)),insert(A),X),bot_bot(set(A))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_3727_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_3728_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_3729_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_3730_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),B3) ).

% Diff_insert2
tff(fact_3731_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),A3)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_3732_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_3733_complex__i__not__neg__numeral,axiom,
    ! [W2: num] : imaginary_unit != aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W2)) ).

% complex_i_not_neg_numeral
tff(fact_3734_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% and_nat_def
tff(fact_3735_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$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)),insert(A),X),B3))
    <=> $ite(aa(set(A),$o,member(A,X),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),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_3736_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B3)
     => aa(set(A),$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)),insert(A),X),B3)) ) ).

% Diff_single_insert
tff(fact_3737_Compl__insert,axiom,
    ! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(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)),insert(A),X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_3738_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3739_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( complex2(X,Y) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3740_complex__of__real__i,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R3)),imaginary_unit) = complex2(zero_zero(real),R3) ).

% complex_of_real_i
tff(fact_3741_i__complex__of__real,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),R3)) = complex2(zero_zero(real),R3) ).

% i_complex_of_real
tff(fact_3742_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$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)),insert(A),X),B3))
    <=> $ite(
          aa(set(A),$o,member(A,X),B3),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3),
          $ite(aa(set(A),$o,member(A,X),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),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_3743_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = $ite(
        ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_3744_and__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = 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)),M)
            & ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na) ))),
        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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_3745_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_minus_ii
tff(fact_3746_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit)),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_3747_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_ii
tff(fact_3748_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_3749_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_3750_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3751_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3752_power2__csqrt,axiom,
    ! [Z2: complex] : aa(nat,complex,power_power(complex,csqrt(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z2 ).

% power2_csqrt
tff(fact_3753_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_3754_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_3755_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_3756_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3757_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_3758_DeMoivre,axiom,
    ! [A2: real,Na: nat] : aa(nat,complex,power_power(complex,cis(A2)),Na) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A2)) ).

% DeMoivre
tff(fact_3759_cis__Arg__unique,axiom,
    ! [Z2: complex,X: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z2) = cis(X) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => ( arg(Z2) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_3760_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z2) = cis(arg(Z2)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z2))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z2)),pi) ) ) ).

% Arg_correct
tff(fact_3761_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_3762_of__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,X)) = csqrt(aa(real,complex,real_Vector_of_real(complex),X)) ) ) ).

% of_real_sqrt
tff(fact_3763_Arg__bounded,axiom,
    ! [Z2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z2))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z2)),pi) ) ).

% Arg_bounded
tff(fact_3764_Arg__def,axiom,
    ! [Z2: complex] :
      arg(Z2) = $ite(Z2 = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_df(complex,fun(real,$o),Z2))) ).

% Arg_def
tff(fact_3765_cis__multiple__2pi,axiom,
    ! [Na: real] :
      ( aa(set(real),$o,member(real,Na),ring_1_Ints(real))
     => ( cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),Na)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_3766_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,real,power_power(real,X),aa(int,nat,nat2,Na)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_3767_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_dg(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_3768_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_3769_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3770_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_3771_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_3772_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_3773_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_3774_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_3775_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_3776_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> aa(set(A),$o,member(A,X),ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_3777_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_3778_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_3779_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_3780_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_3781_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_3782_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W2)) ) ).

% inverse_eq_divide_numeral
tff(fact_3783_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,X))
        <=> ~ aa(set(A),$o,member(A,X),ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_3784_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))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3785_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_3786_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,aa(A,A,uminus_uminus(A),X)),ring_1_Ints(A))
        <=> aa(set(A),$o,member(A,X),ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_3787_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => aa(set(A),$o,member(A,aa(A,A,uminus_uminus(A),A2)),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_3788_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => aa(set(A),$o,member(A,aa(nat,A,power_power(A,A2),Na)),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_3789_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_3790_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_3791_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_3792_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_3793_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_3794_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_3795_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real] :
          ( ( X != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,inverse_inverse(real),X)) = aa(A,A,inverse_inverse(A),aa(real,A,real_Vector_of_real(A),X)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_3796_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),A2)),Na) = aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,A2),Na)) ) ).

% power_inverse
tff(fact_3797_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,member(A,one_one(A)),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_3798_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,member(A,zero_zero(A)),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_3799_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => aa(set(A),$o,member(A,aa(A,A,abs_abs(A),A2)),ring_1_Ints(A)) ) ) ).

% Ints_abs
tff(fact_3800_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(set(A),$o,member(A,aa(nat,A,semiring_1_of_nat(A),Na)),ring_1_Ints(A)) ) ).

% Ints_of_nat
tff(fact_3801_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: num] : aa(set(A),$o,member(A,aa(num,A,numeral_numeral(A),Na)),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_3802_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
          <=> aa(A,$o,P2,X4) )
       => ( ! [X_12: A] : aa(A,$o,P2,X_12)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_3803_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_forall_indirect
tff(fact_3804_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
          <=> aa(A,$o,P2,X4) )
       => ( ? [X_12: A] : aa(A,$o,P2,X_12)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_3805_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_ex_indirect
tff(fact_3806_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_3807_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_3808_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_12: A] : aa(A,$o,P,X_12)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_3809_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_3810_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,minus_minus(A,A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_3811_real__sqrt__inverse,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ).

% real_sqrt_inverse
tff(fact_3812_Ints__of__int,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int] : aa(set(A),$o,member(A,aa(int,A,ring_1_of_int(A),Z2)),ring_1_Ints(A)) ) ).

% Ints_of_int
tff(fact_3813_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A,P: fun(A,$o)] :
          ( aa(set(A),$o,member(A,Q3),ring_1_Ints(A))
         => ( ! [Z: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z))
           => aa(A,$o,P,Q3) ) ) ) ).

% Ints_induct
tff(fact_3814_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A] :
          ( aa(set(A),$o,member(A,Q3),ring_1_Ints(A))
         => ~ ! [Z: int] : Q3 != aa(int,A,ring_1_of_int(A),Z) ) ) ).

% Ints_cases
tff(fact_3815_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_3816_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R3: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R3),real_V7770717601297561774m_norm(A,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R3)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3817_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_3818_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_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_3829_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),Na)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3830_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3831_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_3832_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_3833_divide__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_3834_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( aa(set(A),$o,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_3835_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_dh(nat,A)),X3) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X3)) ) ).

% exp_fdiffs
tff(fact_3836_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_3837_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_3838_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_3839_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_3840_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% inverse_le_1_iff
tff(fact_3841_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_3842_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_3843_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_3844_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_3845_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_3846_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,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,minus_minus(A,B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3847_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3848_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_3849_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( aa(set(A),$o,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_3850_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( dvd_dvd(int,B2,A2)
         => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_3851_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_3852_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_3853_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_3854_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% inverse_less_1_iff
tff(fact_3855_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_3856_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(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,minus_minus(A,A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3857_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] : aa(A,$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)))),X) ) ) ).

% reals_Archimedean
tff(fact_3858_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D5: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D5),E)
         => ( aa(real,$o,P,D5)
           => 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_3859_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D5: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D5),E)
         => ( aa(real,$o,P,D5)
           => 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_3860_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_3861_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),X) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3862_ln__inverse,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_3863_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(set(A),$o,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_3864_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( ( X != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3865_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3866_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,Y),ring_1_Ints(A))
           => ( ( X = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3867_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_di(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_3868_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),pi)) = zero_zero(real) )
    <=> aa(set(real),$o,member(real,X),ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_3869_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] :
              ( aa(nat,$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))),X) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3870_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,Na: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,A,power_power(A,X),aa(nat,nat,minus_minus(nat,Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3871_log__inverse,axiom,
    ! [A2: real,X: 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)),X)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_3872_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = $ite(aa(set(A),$o,member(A,X),ring_1_Ints(A)),zero_zero(A),aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,X))) ) ).

% frac_neg
tff(fact_3873_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))) ).

% exp_plus_inverse_exp
tff(fact_3874_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: A] :
          ( ( archimedean_frac(A,X) = A2 )
        <=> ( aa(set(A),$o,member(A,aa(A,A,minus_minus(A,X),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_3875_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,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(set(A),$o,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_3876_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,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(set(A),$o,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_3877_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => 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),X),aa(real,real,inverse_inverse(real),X))) ) ).

% plus_inverse_ge_2
tff(fact_3878_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(nat,real,power_power(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3879_tan__cot,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),X)) ).

% tan_cot
tff(fact_3880_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( aa(set(real),$o,member(real,Na),ring_1_Ints(real))
     => ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),Na)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_3881_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( aa(set(real),$o,member(real,Na),ring_1_Ints(real))
     => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),Na)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_3882_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_3883_real__le__abs__sinh,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_3884_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3885_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A2 ) )
       => ( fChoice(A,P) = A2 ) ) ) ).

% some_equality
tff(fact_3886_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_cy(A,fun(A,$o),X)) = X ).

% some_eq_trivial
tff(fact_3887_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_3888_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),suminf(A,aTP_Lamp_dj(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_3889_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
        <=> ( ( A2 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3890_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_3891_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
        <=> ( ( X = Y )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_3892_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3893_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_3894_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: real,Y: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,real_V8093663219630862766scaleR(A,X),Y)),Na) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,power_power(real,X),Na)),aa(nat,A,power_power(A,Y),Na)) ) ).

% scaleR_power
tff(fact_3895_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W2: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W2))),A2) ) ).

% scaleR_times
tff(fact_3896_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V2: num,W2: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W2)),aa(num,real,numeral_numeral(real),V2))),A2) ) ).

% inverse_scaleR_times
tff(fact_3897_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V2: num,W2: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W2))),aa(num,real,numeral_numeral(real),V2))),A2) ) ).

% fraction_scaleR_times
tff(fact_3898_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_3899_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
           => ( X = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_3900_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A2: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3901_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ).

% summable_scaleR_right
tff(fact_3902_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A2: A,R3: real] :
          ( aa(A,$o,sums(A,X5),A2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(real,fun(nat,A)),X5),R3)),aa(A,A,real_V8093663219630862766scaleR(A,R3),A2)) ) ) ).

% sums_scaleR_right
tff(fact_3903_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R3),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ) ).

% suminf_scaleR_right
tff(fact_3904_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),X: A] :
          ( summable(real,X5)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ).

% summable_scaleR_left
tff(fact_3905_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A2: real,X: A] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,real),fun(A,fun(nat,A)),X5),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) ) ) ).

% sums_scaleR_left
tff(fact_3906_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scaleR_right_mono
tff(fact_3907_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_3908_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_3909_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_3910_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_3911_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_3912_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(real,$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),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_3913_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V2: real,A2: A] :
          ( ( X = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2)),A2) )
        <=> $ite(V2 = zero_zero(real),X = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V2),X) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_3914_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A2: A,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2)),A2) = X )
        <=> $ite(V2 = zero_zero(real),X = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V2),X)) ) ) ).

% vector_fraction_eq_iff
tff(fact_3915_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),E2)),C2)),D3) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3916_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D3))
        <=> aa(A,$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,minus_minus(real,B2),A2)),E2)),D3)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3917_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),X: A] :
          ( summable(real,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X5)),X) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ) ).

% suminf_scaleR_left
tff(fact_3918_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_3919_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_3920_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3921_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$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),D3)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3922_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3923_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_3924_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3925_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3926_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3927_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_3928_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(real,$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),X)),X) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3929_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% scaleR_2
tff(fact_3930_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,X: A,C2: A,Y: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) = Y )
          <=> ( X = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3931_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,X: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) )
          <=> ( aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3932_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_3933_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_3934_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_3935_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_3936_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_3937_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_3938_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_3939_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_3940_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,X: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3941_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_dm(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_3942_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dn(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_converges
tff(fact_3943_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : sin(A,X3) = suminf(A,aTP_Lamp_dn(A,fun(nat,A),X3)) ) ).

% sin_def
tff(fact_3944_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_do(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_converges
tff(fact_3945_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : cos(A,X3) = suminf(A,aTP_Lamp_do(A,fun(nat,A),X3)) ) ).

% cos_def
tff(fact_3946_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_dp(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_3947_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_dq(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_3948_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_3949_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_3950_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_3951_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_3952_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_3953_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_3954_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_3955_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_3956_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X3 ) ) )
     => ( aa(A,$o,P,A2)
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_3957_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% some_eq_ex
tff(fact_3958_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),A3)
          & aa(A,$o,P,X3) )
     => ( ! [X4: A] :
            ( ( aa(set(A),$o,member(A,X4),A3)
              & aa(A,$o,P,X4) )
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% someI2_bex
tff(fact_3959_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_1: A] : aa(A,$o,P,X_1)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_3960_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_1: A] : aa(A,$o,P,X_1)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_3961_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_3962_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dm(A,fun(nat,A),X)),aa(A,A,exp(A),X)) ) ).

% exp_converges
tff(fact_3963_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : aa(A,A,exp(A),X3) = suminf(A,aTP_Lamp_dm(A,fun(nat,A),X3)) ) ).

% exp_def
tff(fact_3964_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_dr(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_3965_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ds(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_minus_converges
tff(fact_3966_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dt(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_minus_converges
tff(fact_3967_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_1)
     => ( ! [X4: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X4)
           => ? [Y2: A] :
                ( aa(A,$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),X4),Y2) ) )
       => ? [F4: fun(nat,A)] :
          ! [N8: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N8),aa(nat,A,F4,N8))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N8),aa(nat,A,F4,N8)),aa(nat,A,F4,aa(nat,nat,suc,N8))) ) ) ) ).

% dependent_nat_choice
tff(fact_3968_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_3969_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_du(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_3970_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3))),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_3971_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dv(A,fun(nat,A),X)),sinh(A,X)) ) ).

% sinh_converges
tff(fact_3972_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dw(A,fun(nat,A),X)),cosh(A,X)) ) ).

% cosh_converges
tff(fact_3973_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_dy(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_3974_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ea(A,fun(A,fun(nat,A)),X),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_3975_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( sinh(real,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_3976_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,X)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ).

% sinh_real_less_iff
tff(fact_3977_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% sinh_real_le_iff
tff(fact_3978_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_3979_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X) ) ).

% sinh_real_pos_iff
tff(fact_3980_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% sinh_real_nonneg_iff
tff(fact_3981_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_3982_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),aa(A,set(A),set_ord_atMost(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_3983_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_3984_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(set(B),$o,member(B,aa(A,B,F2,X4)),ring_1_Ints(B)) )
         => aa(set(B),$o,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_3985_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_eb(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_3986_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_ec(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_3987_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),F2),S)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ed(fun(B,real),fun(B,A),F2)),S) ) ).

% of_real_sum
tff(fact_3988_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X)),aa(A,set(A),set_ord_atMost(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% atMost_subset_iff
tff(fact_3989_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_3990_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3991_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A3: set(B),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),G),A3)),X) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_ee(fun(B,real),fun(A,fun(B,A)),G),X)),A3) ) ).

% scaleR_left.sum
tff(fact_3992_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F2: fun(B,real),A3: set(B),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),F2),A3)),X) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_ee(fun(B,real),fun(A,fun(B,A)),F2),X)),A3) ) ).

% scaleR_sum_left
tff(fact_3993_cosh__real__nonzero,axiom,
    ! [X: real] : cosh(real,X) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_3994_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ef(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(nat,nat,binomial(aa(nat,nat,suc,Na)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3995_sinh__less__cosh__real,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,X)),cosh(real,X)) ).

% sinh_less_cosh_real
tff(fact_3996_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S2: set(A),F2: fun(A,B),G: fun(A,real)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),S2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(A,real,G,X4)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S2))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),G),S2)) ) ) ).

% sum_norm_le
tff(fact_3997_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),aTP_Lamp_eg(fun(B,A),fun(B,real),F2)),A3)) ) ).

% norm_sum
tff(fact_3998_sinh__le__cosh__real,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),cosh(real,X)) ).

% sinh_le_cosh_real
tff(fact_3999_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_eh(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3),A2) ) ).

% mod_sum_eq
tff(fact_4000_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F2: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ei(real,fun(fun(B,A),fun(B,A)),A2),F2)),A3) ) ).

% scaleR_sum_right
tff(fact_4001_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),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),aa(fun(B,A),fun(B,A),aTP_Lamp_ei(real,fun(fun(B,A),fun(B,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_4002_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => summable(B,aa(A,fun(nat,B),F2,I2)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ek(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) ) ) ).

% summable_sum
tff(fact_4003_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B)),X: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F2,I2)),aa(A,B,X,I2)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ek(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5)) ) ) ).

% sums_sum
tff(fact_4004_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_4005_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_em(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_4006_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,D3: fun(nat,A)] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),D3),X2)),aa(nat,set(nat),set_ord_atMost(nat),Na))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D3,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4007_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_4008_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_eo(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_4009_sum__choose__lower,axiom,
    ! [R3: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ep(nat,fun(nat,nat),R3)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R3),Na))),Na) ).

% sum_choose_lower
tff(fact_4010_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_eq(nat,fun(nat,nat),Na)),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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),Na),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_4011_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_eq(nat,fun(nat,nat),Na)),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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),Na),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_4012_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Na: nat,K: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),C2),W)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4013_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4014_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_es(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = 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),Na))),one_one(A))),Na) ) ).

% gbinomial_parallel_sum
tff(fact_4015_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_et(nat,fun(nat,fun(nat,nat)),M),Na)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_4016_vandermonde,axiom,
    ! [M: nat,Na: nat,R3: 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_eu(nat,fun(nat,fun(nat,fun(nat,nat))),M),Na),R3)),aa(nat,set(nat),set_ord_atMost(nat),R3)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),R3) ).

% vandermonde
tff(fact_4017_cosh__real__pos,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,X)) ).

% cosh_real_pos
tff(fact_4018_cosh__real__nonneg,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,X)) ).

% cosh_real_nonneg
tff(fact_4019_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4020_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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_eq(real),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4021_cosh__real__ge__1,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,X)) ).

% cosh_real_ge_1
tff(fact_4022_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sinh(A,X))),cosh(A,X)) ) ).

% sinh_double
tff(fact_4023_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)),insert(nat),aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_4024_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))) ) ).

% sum_gp_basic
tff(fact_4025_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => summable(B,aa(A,fun(nat,B),F2,I2)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ek(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_ev(fun(A,fun(nat,B)),fun(A,B),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_4026_choose__row__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% choose_row_sum
tff(fact_4027_binomial,axiom,
    ! [A2: nat,B2: nat,Na: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Na) = 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_ew(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ).

% binomial
tff(fact_4028_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4029_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4030_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% sum.in_pairs_0
tff(fact_4031_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Na: nat,B2: fun(nat,A),X: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),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),X)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_fd(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_4032_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fe(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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))),M)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4033_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Na) = 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_ff(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% binomial_ring
tff(fact_4034_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Na: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),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_fg(fun(nat,nat),fun(nat,fun(nat,nat)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fg(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fi(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_4035_choose__square__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fj(nat,fun(nat,nat),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Na) ).

% choose_square_sum
tff(fact_4036_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Na) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% pochhammer_binomial_sum
tff(fact_4037_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,X)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4038_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( 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,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4039_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),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,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4040_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),B2))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4041_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cosh(A,X)),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,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4042_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4043_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4044_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)),insert(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_4045_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fl(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),I5)) ) ).

% sum_power_add
tff(fact_4046_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_fm(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_fn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4047_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,real,arcosh(real),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_4048_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: 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_fo(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fp(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4049_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),Na)) = 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_fq(A,fun(A,fun(nat,fun(nat,A))),X),Y),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4050_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_4051_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z2: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( ( aa(nat,A,power_power(A,Z2),Na) = 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_fr(nat,fun(A,fun(A,fun(nat,A))),Na),Z2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4052_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na)))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp0
tff(fact_4053_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( ( Na != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fs(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4054_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ft(nat,fun(nat,A),M)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4055_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: 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_fo(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fu(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4056_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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))),M)) ).

% binomial_r_part_sum
tff(fact_4057_choose__linear__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fv(nat,fun(nat,nat),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% choose_linear_sum
tff(fact_4058_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fw(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4059_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Na: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M8: real] :
            ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z3))
             => aa(real,$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_bi(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Na)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4060_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_4061_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fx(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4062_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fy(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_even_sum
tff(fact_4063_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fz(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_odd_sum
tff(fact_4064_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4065_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Na))) ) ).

% mask_eq_sum_exp
tff(fact_4066_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> aa(set(A),$o,member(A,aa(A,A,exp(A),X)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_4067_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z2: A] : cosh(A,Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z2)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_4068_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_4069_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z2: A] : sinh(A,Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Z2)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_4070_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,power_power(A,aa(A,A,exp(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4071_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cosh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% cosh_def
tff(fact_4072_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_4073_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sinh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,minus_minus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% sinh_def
tff(fact_4074_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_4075_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gc(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_4076_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_gd(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_4077_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_gd(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs
tff(fact_4078_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_4079_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,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_ge(fun(A,B),fun(fun(A,B),fun(A,B)),X),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4080_of__nat__id,axiom,
    ! [Na: nat] : aa(nat,nat,semiring_1_of_nat(nat),Na) = Na ).

% of_nat_id
tff(fact_4081_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_gf(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_4082_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gd(fun(B,A),fun(B,A),F2)),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gd(fun(B,A),fun(B,A),F2)),A3) ) ).

% abs_sum_abs
tff(fact_4083_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S: set(A)] : aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),aTP_Lamp_gg(fun(A,real),fun(A,complex),F2)),S) = complex2(aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),F2),S),zero_zero(real)) ).

% Complex_sum'
tff(fact_4084_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_gh(fun(A,nat),fun(A,int),F2)),A3) ).

% int_sum
tff(fact_4085_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X4)),aa(A,nat,F2,X4)) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_gi(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(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_4086_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Na: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,Na) )
     => ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X4)) ) ) ).

% sum_SucD
tff(fact_4087_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),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(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_4088_sum__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_gj(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_gk(nat,fun(complex,fun(complex,$o)),Na),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4089_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_gl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(set(B),fun(C,A),aTP_Lamp_gn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B3) ) ).

% sum.swap
tff(fact_4090_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_gj(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_go(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4091_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( aa(set(B),$o,member(B,A4),A3)
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4092_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => ( aa(A,B,G,X4) = 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_4093_dvd__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),D3: B,F2: fun(A,B)] :
          ( ! [A4: A] :
              ( aa(set(A),$o,member(A,A4),A3)
             => dvd_dvd(B,D3,aa(A,B,F2,A4)) )
         => dvd_dvd(B,D3,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% dvd_sum
tff(fact_4094_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K6: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),K6)
             => 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),K6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),K6)) ) ) ).

% sum_mono
tff(fact_4095_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_gq(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3)),A3) ) ).

% sum_product
tff(fact_4096_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),R3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_gr(fun(B,A),fun(A,fun(B,A)),F2),R3)),A3) ) ).

% sum_distrib_right
tff(fact_4097_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R3: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gs(A,fun(fun(B,A),fun(B,A)),R3),F2)),A3) ) ).

% sum_distrib_left
tff(fact_4098_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_gt(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_4099_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gu(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,minus_minus(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ).

% sum_subtractf
tff(fact_4100_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),R3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_gv(fun(B,A),fun(A,fun(B,A)),F2),R3)),A3) ) ).

% sum_divide_distrib
tff(fact_4101_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gw(fun(B,A),fun(B,A),F2)),A3) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) ) ).

% sum_negf
tff(fact_4102_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => 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_4103_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),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_4104_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3)
         => ( ! [X4: nat] :
                ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,X4)),A3)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X4)) = aa(nat,A,G,aa(nat,nat,suc,X4)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_4105_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),zero_zero(real))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gx(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_4106_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),X)
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gx(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4107_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),X)
            & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gy(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4108_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),X)
          & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gy(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4109_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),aa(A,set(A),set_ord_lessThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_4110_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X)),aa(A,set(A),set_ord_lessThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% lessThan_subset_iff
tff(fact_4111_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_4112_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(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)),insert(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_4113_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gz(nat,real)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4114_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ha(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_4115_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),Na))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na) ) ) ).

% lessThan_strict_subset_iff
tff(fact_4116_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),Na) = bot_bot(set(nat)) )
    <=> ( Na = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_4117_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)),insert(nat),K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_4118_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_4119_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)),insert(nat),pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_4120_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hb(fun(nat,A),fun(nat,fun(nat,A)),G),Na)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = 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),Na)) ) ).

% sum.nat_diff_reindex
tff(fact_4121_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Na: A] :
          ( ! [X4: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X4)),aa(A,nat,P,X4))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),Na))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),Na))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_hc(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_4122_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),N))),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),X) ) ) ) ).

% suminf_le_const
tff(fact_4123_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_4124_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hd(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_4125_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_em(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_4126_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Na: nat,R3: A] : aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),R3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_he(fun(nat,A),fun(A,fun(nat,A)),F2),R3)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sumr_diff_mult_const2
tff(fact_4127_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N: nat] : aa(A,$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))),X)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_4128_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),S)
        <=> aa(A,$o,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),Na)))) ) ) ).

% sums_iff_shift
tff(fact_4129_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(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),Na))))
        <=> aa(A,$o,sums(A,F2),S) ) ) ).

% sums_iff_shift'
tff(fact_4130_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Na: nat] :
          ( aa(A,$o,sums(A,F2),S)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(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),Na)))) ) ) ).

% sums_split_initial_segment
tff(fact_4131_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% power_diff_1_eq
tff(fact_4132_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% one_diff_power_eq
tff(fact_4133_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Na: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_4134_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% sum.atMost_shift
tff(fact_4135_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_aw(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_4136_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_aw(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(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_4137_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Na))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_4138_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp_strict
tff(fact_4139_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z2: A,H: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hf(A,fun(A,fun(nat,fun(nat,A))),Z2),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hg(A,fun(A,fun(nat,fun(nat,A))),Z2),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) ) ).

% lemma_termdiff1
tff(fact_4140_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))),aa(nat,A,power_power(A,Y),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hh(A,fun(nat,fun(A,fun(nat,A))),X),Na),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_4141_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hi(A,fun(nat,fun(A,fun(nat,A))),X),Na),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% power_diff_sumr2
tff(fact_4142_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Na: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = 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),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4143_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Na: 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),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,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),Na)))),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),Na))) ) ).

% polyfun_linear_factor
tff(fact_4144_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K6) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),K6)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_4145_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Na))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_4146_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hj(A,fun(nat,fun(nat,A)),X),Na)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% one_diff_power_eq'
tff(fact_4147_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,Na: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( Na != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_hk(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_4148_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B5: 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_hl(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),B5),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,H),Na)),semiring_char_0_fact(real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_4149_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),Na: 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_hm(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))),Na))) = 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_hn(fun(nat,real),fun(nat,real),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ho(fun(nat,real),fun(nat,real),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ).

% sum_split_even_odd
tff(fact_4150_Maclaurin__exp__le,axiom,
    ! [X: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
      & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_hp(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_4151_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_hr(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4152_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,K: nat] : aa(A,A,exp(A),X) = 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_dm(A,fun(nat,A),X)),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_hs(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4153_Maclaurin__sin__bound,axiom,
    ! [X: real,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,X)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gy(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),X)),Na))) ).

% Maclaurin_sin_bound
tff(fact_4154_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D5: 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)))),D5)))),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)))),D5)),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_4155_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_hp(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4156_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z2: A,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H)),Na)),aa(nat,A,power_power(A,Z2),Na))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z2),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hu(A,fun(A,fun(nat,fun(nat,A))),H),Z2),Na)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_4157_Maclaurin__sin__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T4: real] : sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gy(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ).

% Maclaurin_sin_expansion
tff(fact_4158_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
      & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gy(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4159_Maclaurin__cos__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
      & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gx(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4160_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => bij_betw(nat,complex,aTP_Lamp_hv(nat,fun(nat,complex),Na),aa(nat,set(nat),set_ord_lessThan(nat),Na),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_go(nat,fun(complex,$o),Na))) ) ).

% bij_betw_roots_unity
tff(fact_4161_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),
            zero_zero(A),
            $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na)))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ) ).

% sum_gp
tff(fact_4162_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fx(A,fun(nat,A),R3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4163_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_4164_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_4165_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H2) )
        <=> ( ( ( L = L3 )
              & ( H = H2 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L3),H2) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_4166_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_4167_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_4168_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4169_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_4170_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)),insert(A),C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4171_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_4172_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_4173_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% sum.cl_ivl_Suc
tff(fact_4174_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S2,T5)
         => ( 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_hw(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),T5) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_4175_bij__betwI_H,axiom,
    ! [A: $tType,B: $tType,X5: set(A),F2: fun(A,B),Y4: set(B)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),X5)
         => ! [Y3: A] :
              ( aa(set(A),$o,member(A,Y3),X5)
             => ( ( aa(A,B,F2,X4) = aa(A,B,F2,Y3) )
              <=> ( X4 = Y3 ) ) ) )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),X5)
           => aa(set(B),$o,member(B,aa(A,B,F2,X4)),Y4) )
       => ( ! [Y3: B] :
              ( aa(set(B),$o,member(B,Y3),Y4)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),X5)
                  & ( Y3 = aa(A,B,F2,X3) ) ) )
         => bij_betw(A,B,F2,X5,Y4) ) ) ) ).

% bij_betwI'
tff(fact_4176_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4177_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A,H2: A] : ~ aa(set(A),$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,L3,H2)) ) ).

% not_Iic_le_Icc
tff(fact_4178_ex__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Na)
          & aa(nat,$o,P,M3) )
    <=> ? [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X2) ) ) ).

% ex_nat_less
tff(fact_4179_all__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Na)
         => aa(nat,$o,P,M3) )
    <=> ! [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X2) ) ) ).

% all_nat_less
tff(fact_4180_atMost__atLeast0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),Na) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Na) ).

% atMost_atLeast0
tff(fact_4181_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4182_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hx(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_4183_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4184_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_4185_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = set_or1337092689740270186AtMost(nat,M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_4186_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4187_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4188_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Na,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4189_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_4190_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_4191_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_4192_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_4193_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4194_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hd(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,aa(nat,nat,suc,Na))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_4195_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4196_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_4197_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hz(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sum.nested_swap'
tff(fact_4198_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_ic(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4199_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4200_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_atMost(nat),Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4201_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)))) ) ).

% sum_up_index_split
tff(fact_4202_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_id(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_4203_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ie(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_4204_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4205_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,$o),ord_less(real),zero_zero(real)),E2)
           => ~ ! [N7: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M2)
                     => ! [N8: nat] : aa(real,$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,M2,N8)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4206_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))) ) ) ) ).

% sum_gp_multiplied
tff(fact_4207_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.in_pairs
tff(fact_4208_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,K: A] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X2: nat] :
                ( aa(set(nat),$o,member(nat,X2),set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C2,X2) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4209_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_if(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4210_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ig(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_4211_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_4212_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D3: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ih(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D3))) ) ).

% double_arith_series
tff(fact_4213_arith__series__nat,axiom,
    ! [A2: nat,D3: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ii(nat,fun(nat,fun(nat,nat)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),D3)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_4214_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ig(nat,nat)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_4215_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_4216_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D3: A,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ij(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D3)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_4217_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_4218_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = $ite(X = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp_offset
tff(fact_4219_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_il(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ) ) ).

% polyfun_diff
tff(fact_4220_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_im(A,fun(nat,A),Z2),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_4221_VEBT__internal_Oinrange,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% VEBT_internal.inrange
tff(fact_4222_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Na)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_in(int,int)),set_or1337092689740270186AtMost(int,M,Na)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_4223_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),Na),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% push_bit_numeral_minus_1
tff(fact_4224_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4730199178511100633sh_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_4225_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4730199178511100633sh_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_4226_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se4730199178511100633sh_bit(A,Na,zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_4227_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit(A,Na,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4228_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,M,bit_se4730199178511100633sh_bit(A,Na,A2)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),A2) ) ).

% push_bit_push_bit
tff(fact_4229_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(set(B),$o,member(B,aa(A,B,F2,X4)),ring_1_Ints(B)) )
         => aa(set(B),$o,member(B,groups7121269368397514597t_prod(A,B,F2,A3)),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_4230_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4730199178511100633sh_bit(A,Na,A2)),bit_se4730199178511100633sh_bit(A,Na,B2)) ) ).

% push_bit_and
tff(fact_4231_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4730199178511100633sh_bit(A,Na,A2)),bit_se4730199178511100633sh_bit(A,Na,B2)) ) ).

% push_bit_xor
tff(fact_4232_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : aa(int,int,bit_concat_bit(Na,zero_zero(int)),L) = bit_se4730199178511100633sh_bit(int,Na,L) ).

% concat_bit_of_zero_1
tff(fact_4233_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_io(B,A),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_4234_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_ip(fun(B,nat),fun(B,A),F2),A3) ) ).

% of_nat_prod
tff(fact_4235_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_iq(fun(B,int),fun(B,A),F2),A3) ) ).

% of_int_prod
tff(fact_4236_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F2: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,F2,S)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_ir(fun(B,real),fun(B,A),F2),S) ) ).

% of_real_prod
tff(fact_4237_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Na),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,Na,aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_4238_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Na),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,Na,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_4239_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_numeral
tff(fact_4240_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_4241_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : bit_se4730199178511100633sh_bit(nat,Na,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% push_bit_of_Suc_0
tff(fact_4242_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Na),A2) = bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_4243_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se4730199178511100633sh_bit(A,Na,one_one(A)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% push_bit_of_1
tff(fact_4244_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se4730199178511100633sh_bit(A,Na,A2))
        <=> ( ( Na != zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_4245_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% prod.cl_ivl_Suc
tff(fact_4246_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_minus_numeral
tff(fact_4247_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S2,T5)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_is(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S2) = groups7121269368397514597t_prod(B,C,G,T5) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_4248_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_it(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,H,A3)) ) ).

% prod.distrib
tff(fact_4249_prod__dvd__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [A4: A] :
              ( aa(set(A),$o,member(A,A4),A3)
             => dvd_dvd(B,aa(A,B,F2,A4),aa(A,B,G,A4)) )
         => dvd_dvd(B,groups7121269368397514597t_prod(A,B,F2,A3),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ).

% prod_dvd_prod
tff(fact_4250_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_iu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3),A3) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_iw(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3),B3) ) ).

% prod.swap
tff(fact_4251_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,M,Na)) = bit_se4730199178511100633sh_bit(A,M,aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_push_bit
tff(fact_4252_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : bit_se4730199178511100633sh_bit(A,Na,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,Na,M)) ) ).

% push_bit_of_nat
tff(fact_4253_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: set(B),Na: nat] : aa(nat,A,power_power(A,groups7121269368397514597t_prod(B,A,F2,A3)),Na) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_ix(fun(B,A),fun(nat,fun(B,A)),F2),Na),A3) ) ).

% prod_power_distrib
tff(fact_4254_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_iy(fun(B,A),fun(B,A),F2),A3) ) ).

% abs_prod
tff(fact_4255_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_iz(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_4256_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_eh(fun(B,A),fun(A,fun(B,A)),F2),A2),A3),A2) = modulo_modulo(A,groups7121269368397514597t_prod(B,A,F2,A3),A2) ) ).

% mod_prod_eq
tff(fact_4257_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),bit_se4730199178511100633sh_bit(A,Na,A2)) ) ).

% push_bit_minus
tff(fact_4258_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,Na,A2)),bit_se4730199178511100633sh_bit(A,Na,B2)) ) ).

% push_bit_add
tff(fact_4259_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F2: fun(A,B),A3: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_ja(fun(A,B),fun(A,real),F2),A3) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F2,A3)) ) ).

% prod_norm
tff(fact_4260_push__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : bit_se4730199178511100633sh_bit(nat,Na,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se4730199178511100633sh_bit(int,Na,K)) ).

% push_bit_nat_eq
tff(fact_4261_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: int] : bit_se4730199178511100633sh_bit(A,Na,aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),bit_se4730199178511100633sh_bit(int,Na,K)) ) ).

% push_bit_of_int
tff(fact_4262_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A3))),groups7121269368397514597t_prod(B,real,aTP_Lamp_jb(fun(B,A),fun(B,real),F2),A3)) ) ).

% norm_prod_le
tff(fact_4263_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_nonneg
tff(fact_4264_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,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),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ).

% prod_mono
tff(fact_4265_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_pos
tff(fact_4266_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_ge_1
tff(fact_4267_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),bit_se4730199178511100633sh_bit(A,M,A2)) ) ).

% push_bit_take_bit
tff(fact_4268_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4730199178511100633sh_bit(A,Na,A2)) = bit_se4730199178511100633sh_bit(A,Na,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),Na)),A2)) ) ).

% take_bit_push_bit
tff(fact_4269_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4270_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: fun(B,nat),A3: set(B)] : aa(nat,A,power_power(A,C2),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A3)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_jd(A,fun(fun(B,nat),fun(B,A)),C2),F2),A3) ) ).

% power_sum
tff(fact_4271_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_4272_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : bit_se8732182000553998342ip_bit(nat,M,Na) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% flip_bit_nat_def
tff(fact_4273_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_4274_aset_I2_J,axiom,
    ! [D: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)) ) ) ) ) ) ).

% aset(2)
tff(fact_4275_aset_I1_J,axiom,
    ! [D: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)) ) ) ) ) ) ).

% aset(1)
tff(fact_4276_bset_I2_J,axiom,
    ! [D: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D)) ) ) ) ) ) ).

% bset(2)
tff(fact_4277_bset_I1_J,axiom,
    ! [D: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D)) ) ) ) ) ) ).

% bset(1)
tff(fact_4278_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_4279_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),G),Na),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = groups7121269368397514597t_prod(nat,A,G,aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.nat_diff_reindex
tff(fact_4280_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4281_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q3)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q3),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4282_concat__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),bit_se4730199178511100633sh_bit(int,Na,L)) ).

% concat_bit_eq
tff(fact_4283_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se8732182000553998342ip_bit(A,Na,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),bit_se4730199178511100633sh_bit(A,Na,one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_4284_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se8732182000553998342ip_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),bit_se4730199178511100633sh_bit(int,Na,one_one(int))) ).

% flip_bit_int_def
tff(fact_4285_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_4286_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4287_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Na))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4288_aset_I10_J,axiom,
    ! [D3: int,D: int,A3: set(int),T2: int] :
      ( dvd_dvd(int,D3,D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2))
           => ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)),T2)) ) ) ) ).

% aset(10)
tff(fact_4289_aset_I9_J,axiom,
    ! [D3: int,D: int,A3: set(int),T2: int] :
      ( dvd_dvd(int,D3,D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2))
           => dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)),T2)) ) ) ) ).

% aset(9)
tff(fact_4290_bset_I10_J,axiom,
    ! [D3: int,D: int,B3: set(int),T2: int] :
      ( dvd_dvd(int,D3,D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2))
           => ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D)),T2)) ) ) ) ).

% bset(10)
tff(fact_4291_bset_I9_J,axiom,
    ! [D3: int,D: int,B3: set(int),T2: int] :
      ( dvd_dvd(int,D3,D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2))
           => dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D)),T2)) ) ) ) ).

% bset(9)
tff(fact_4292_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,Na,A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_4293_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4294_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4295_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_4296_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4297_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4298_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ig(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Na))) ) ).

% fact_prod
tff(fact_4299_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_jh(fun(nat,fun(nat,A)),fun(nat,A),A2),aa(nat,set(nat),set_ord_atMost(nat),Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.nested_swap'
tff(fact_4300_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se4730199178511100633sh_bit(A,Na,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_4301_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_jk(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4302_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] : bit_se4730199178511100633sh_bit(A,M,bit_se2239418461657761734s_mask(A,Na)) = 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),Na),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_4303_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se4730199178511100633sh_bit(A,Na,one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_4304_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,Na,one_one(int)))) ).

% unset_bit_int_def
tff(fact_4305_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_4306_periodic__finite__ex,axiom,
    ! [D3: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P,X4)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X2: int] :
              ( aa(set(int),$o,member(int,X2),set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,P,X2) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4307_aset_I7_J,axiom,
    ! [D: int,A3: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)) ) ) ) ).

% aset(7)
tff(fact_4308_aset_I5_J,axiom,
    ! [D: int,T2: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,T2),A3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)),T2) ) ) ) ) ).

% aset(5)
tff(fact_4309_aset_I4_J,axiom,
    ! [D: int,T2: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,T2),A3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_4310_aset_I3_J,axiom,
    ! [D: int,T2: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_4311_bset_I7_J,axiom,
    ! [D: int,T2: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,T2),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,minus_minus(int,X3),D)) ) ) ) ) ).

% bset(7)
tff(fact_4312_bset_I5_J,axiom,
    ! [D: int,B3: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X3),D)),T2) ) ) ) ).

% bset(5)
tff(fact_4313_bset_I4_J,axiom,
    ! [D: int,T2: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,T2),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,minus_minus(int,X3),D) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_4314_bset_I3_J,axiom,
    ! [D: int,T2: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,aa(int,int,minus_minus(int,T2),one_one(int))),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,minus_minus(int,X3),D) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_4315_push__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : bit_se4730199178511100633sh_bit(nat,Na,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% push_bit_nat_def
tff(fact_4316_push__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se4730199178511100633sh_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_int_def
tff(fact_4317_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z2: fun(A,B),W2: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z2,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W2,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,groups7121269368397514597t_prod(A,B,Z2,I5)),groups7121269368397514597t_prod(A,B,W2,I5)))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aa(fun(A,B),fun(A,real),aTP_Lamp_jl(fun(A,B),fun(fun(A,B),fun(A,real)),Z2),W2)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_4318_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% prod.atMost_shift
tff(fact_4319_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Na,A2) = 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))),Na)) ) ).

% push_bit_eq_mult
tff(fact_4320_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na),A2)
         => ~ ! [B4: A] : A2 != bit_se4730199178511100633sh_bit(A,Na,B4) ) ) ).

% exp_dvdE
tff(fact_4321_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Na)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ig(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4322_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_4323_aset_I8_J,axiom,
    ! [D: int,A3: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)) ) ) ) ).

% aset(8)
tff(fact_4324_aset_I6_J,axiom,
    ! [D: int,T2: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D)),T2) ) ) ) ) ).

% aset(6)
tff(fact_4325_bset_I8_J,axiom,
    ! [D: int,T2: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( aa(set(int),$o,member(int,aa(int,int,minus_minus(int,T2),one_one(int))),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,minus_minus(int,X3),D)) ) ) ) ) ).

% bset(8)
tff(fact_4326_bset_I6_J,axiom,
    ! [D: int,B3: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,X3),D)),T2) ) ) ) ).

% bset(6)
tff(fact_4327_cpmi,axiom,
    ! [D: 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)),D)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Z3)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P2,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa3: int] :
                  ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),B3)
                     => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D)) ) )
         => ( ! [X4: int,K2: int] :
                ( aa(int,$o,P2,X4)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X2: int] :
                    ( aa(set(int),$o,member(int,X2),set_or1337092689740270186AtMost(int,one_one(int),D))
                    & aa(int,$o,P2,X2) )
                | ? [X2: int] :
                    ( aa(set(int),$o,member(int,X2),set_or1337092689740270186AtMost(int,one_one(int),D))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),B3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X2)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4328_cppi,axiom,
    ! [D: 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)),D)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X4)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P2,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa3: int] :
                  ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),A3)
                     => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D)) ) )
         => ( ! [X4: int,K2: int] :
                ( aa(int,$o,P2,X4)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X2: int] :
                    ( aa(set(int),$o,member(int,X2),set_or1337092689740270186AtMost(int,one_one(int),D))
                    & aa(int,$o,P2,X2) )
                | ? [X2: int] :
                    ( aa(set(int),$o,member(int,X2),set_or1337092689740270186AtMost(int,one_one(int),D))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),A3)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa2),X2)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4329_push__bit__minus__one,axiom,
    ! [Na: nat] : bit_se4730199178511100633sh_bit(int,Na,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_minus_one
tff(fact_4330_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jm(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_4331_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,Na) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jn(A,fun(nat,fun(nat,A)),A2),Na),set_or1337092689740270186AtMost(nat,one_one(nat),Na)) ) ).

% pochhammer_prod_rev
tff(fact_4332_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ig(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4333_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jo(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.in_pairs
tff(fact_4334_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : 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))),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jo(fun(nat,A),fun(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% prod.in_pairs_0
tff(fact_4335_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jn(A,fun(nat,fun(nat,A)),A2),Na),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4336_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)
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jp(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),aa(nat,set(nat),set_ord_atMost(nat),P3)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4337_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),A2),
            $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Na),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_4338_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jr(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_4339_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,root(Na),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),Na))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_go(nat,fun(complex,$o),Na)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_js(complex,fun(nat,fun(complex,$o)),C2),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4340_VEBT__internal_Oinsert__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_VEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) = vEBT_VEBT_set_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)) ) ) ) ).

% VEBT_internal.insert_corr
tff(fact_4341_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_4342_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(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_4343_UnCI,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( ( ~ aa(set(A),$o,member(A,C2),B3)
       => aa(set(A),$o,member(A,C2),A3) )
     => aa(set(A),$o,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_4344_Un__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( aa(set(A),$o,member(A,C2),A3)
        | aa(set(A),$o,member(A,C2),B3) ) ) ).

% Un_iff
tff(fact_4345_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_4346_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3) ) ) ).

% Un_subset_iff
tff(fact_4347_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)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_insert_right
tff(fact_4348_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_insert_left
tff(fact_4349_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),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_4350_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),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_4351_real__root__zero,axiom,
    ! [Na: nat] : aa(real,real,root(Na),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4352_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_4353_real__root__eq__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = aa(real,real,root(Na),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4354_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_4355_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),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_4356_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_4357_real__root__eq__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4358_real__root__less__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ).

% real_root_less_iff
tff(fact_4359_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ).

% real_root_le_iff
tff(fact_4360_real__root__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4361_real__root__eq__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4362_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_4363_real__root__lt__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_4364_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_4365_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_4366_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_4367_real__root__lt__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),X)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_4368_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_4369_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_4370_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4371_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(A,nat,F2,A3)) = groups7121269368397514597t_prod(A,int,aTP_Lamp_gh(fun(A,nat),fun(A,int),F2),A3) ).

% int_prod
tff(fact_4372_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),C3)),aa(set(A),set(A),minus_minus(set(A),B3),C3)) ).

% Un_Diff
tff(fact_4373_insert__def,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cy(A,fun(A,$o),A2))),B3) ).

% insert_def
tff(fact_4374_real__root__inverse,axiom,
    ! [Na: nat,X: real] : aa(real,real,root(Na),aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Na),X)) ).

% real_root_inverse
tff(fact_4375_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_4376_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_4377_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D)
       => aa(set(A),$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)),C3),D)) ) ) ).

% Un_mono
tff(fact_4378_Un__least,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$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)),C3) ) ) ).

% Un_least
tff(fact_4379_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_4380_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_4381_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_4382_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_4383_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ~ ! [A6: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),A3)
           => ! [B8: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),B3)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B8) ) ) ) ) ).

% subset_UnE
tff(fact_4384_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_4385_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,X: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),X) = aa(real,real,root(M),aa(real,real,root(Na),X)) ).

% real_root_mult_exp
tff(fact_4386_real__root__mult,axiom,
    ! [Na: nat,X: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ).

% real_root_mult
tff(fact_4387_Un__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jt(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Un_def
tff(fact_4388_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ju(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_4389_UnE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ( ~ aa(set(A),$o,member(A,C2),A3)
       => aa(set(A),$o,member(A,C2),B3) ) ) ).

% UnE
tff(fact_4390_UnI1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),A3)
     => aa(set(A),$o,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_4391_UnI2,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,member(A,C2),B3)
     => aa(set(A),$o,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_4392_bex__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
          & aa(A,$o,P,X2) )
    <=> ( ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
            & aa(A,$o,P,X2) )
        | ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),B3)
            & aa(A,$o,P,X2) ) ) ) ).

% bex_Un
tff(fact_4393_ball__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => aa(A,$o,P,X2) )
    <=> ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => aa(A,$o,P,X2) )
        & ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),B3)
           => aa(A,$o,P,X2) ) ) ) ).

% ball_Un
tff(fact_4394_Un__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_assoc
tff(fact_4395_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_4396_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_4397_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_4398_Un__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_left_commute
tff(fact_4399_real__root__commute,axiom,
    ! [M: nat,Na: nat,X: real] : aa(real,real,root(M),aa(real,real,root(Na),X)) = aa(real,real,root(Na),aa(real,real,root(M),X)) ).

% real_root_commute
tff(fact_4400_real__root__divide,axiom,
    ! [Na: nat,X: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ).

% real_root_divide
tff(fact_4401_real__root__minus,axiom,
    ! [Na: nat,X: real] : aa(real,real,root(Na),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Na),X)) ).

% real_root_minus
tff(fact_4402_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_jv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_4403_real__root__pos__pos__le,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),X)) ) ).

% real_root_pos_pos_le
tff(fact_4404_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4405_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_4406_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4407_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4408_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),C3)
    <=> aa(set(A),$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),C3)) ) ).

% Diff_subset_conv
tff(fact_4409_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),minus_minus(set(A),B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_4410_real__root__less__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_4411_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_4412_real__root__power,axiom,
    ! [Na: nat,X: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,root(Na),X)),K) ) ) ).

% real_root_power
tff(fact_4413_real__root__abs,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(Na),X)) ) ) ).

% real_root_abs
tff(fact_4414_sgn__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Na),X)) = aa(real,real,sgn_sgn(real),X) ) ) ).

% sgn_root
tff(fact_4415_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,J)) = groups7121269368397514597t_prod(int,int,aTP_Lamp_in(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_4416_real__root__gt__zero,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),X)) ) ) ).

% real_root_gt_zero
tff(fact_4417_real__root__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N3),X)),aa(real,real,root(Na),X)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4418_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_4419_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Na),aa(nat,real,power_power(real,Y),Na))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4420_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_4421_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_4422_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_in(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_4423_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_4424_real__root__strict__increasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),X)),aa(real,real,root(N3),X)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4425_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N3),X)),aa(real,real,root(Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_4426_odd__real__root__pow,axiom,
    ! [Na: nat,X: real] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ).

% odd_real_root_pow
tff(fact_4427_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( ( aa(nat,real,power_power(real,Y),Na) = X )
       => ( aa(real,real,root(Na),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_4428_odd__real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),Na)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_4429_real__root__pow__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4430_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4431_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,power_power(real,Y),Na) = X )
         => ( aa(real,real,root(Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4432_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(N3),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_4433_sgn__power__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(Na),X))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),aa(real,real,root(Na),X))),Na)) = X ) ) ).

% sgn_power_root
tff(fact_4434_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_4435_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N2)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_4436_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M4)),aa(nat,A,X5,N)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_4437_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
               => ! [N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N8)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M2)),aa(nat,A,X5,N8)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_4438_log__root,axiom,
    ! [Na: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log(B2),aa(real,real,root(Na),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_4439_log__base__root,axiom,
    ! [Na: nat,B2: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Na),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_4440_ln__root,axiom,
    ! [Na: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_4441_split__root,axiom,
    ! [P: fun(real,$o),Na: nat,X: real] :
      ( aa(real,$o,P,aa(real,real,root(Na),X))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y5)),Na)) = X )
             => aa(real,$o,P,Y5) ) ) ) ) ).

% split_root
tff(fact_4442_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,real,root(Na),X) = powr(real,X,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ) ).

% root_powr_inverse
tff(fact_4443_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_4444_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% le_sup_iff
tff(fact_4445_set__vebt__insert,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n))
       => ( vEBT_set_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ) ).

% set_vebt_insert
tff(fact_4446_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A7: set(B)] :
      ( ~ aa(set(A),$o,member(A,B2),A3)
     => ( ~ aa(set(B),$o,member(B,aa(A,B,F2,B2)),A7)
       => ( bij_betw(A,B,F2,A3,A7)
        <=> 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)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A7),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_4447_set__vebt__buildup,axiom,
    ! [I: nat] : vEBT_set_vebt(vEBT_vebt_buildup(I)) = bot_bot(set(nat)) ).

% set_vebt_buildup
tff(fact_4448_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,T_2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T_1,n)
     => ( vEBT_invar_vebt(T_2,n)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% set_vebt_equal
tff(fact_4449_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),B3))) ).

% sup_set_def
tff(fact_4450_sup__Un__eq,axiom,
    ! [A: $tType,R2: set(A),S2: set(A),X3: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),R2)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R2),S2)) ) ).

% sup_Un_eq
tff(fact_4451_vebt__inst_Oset__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,Na: nat,T_2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T_1,Na)
     => ( vEBT_invar_vebt(T_2,Na)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% vebt_inst.set_vebt_equal
tff(fact_4452_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(4)
tff(fact_4453_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(3)
tff(fact_4454_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X) ) ) ) ).

% le_supE
tff(fact_4455_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X) ) ) ) ).

% le_supI
tff(fact_4456_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge1
tff(fact_4457_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge2
tff(fact_4458_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_4459_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_4460_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_4461_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)) ) ) ) ).

% sup_mono
tff(fact_4462_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) ) ) ) ).

% sup_least
tff(fact_4463_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_4464_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_4465_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_4466_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,X4),Y3))
         => ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F2,X4),Y3))
           => ( ! [X4: A,Y3: A,Z: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y3),Z)),X4) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_4467_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_4468_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_4469_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_4470_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_4471_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_4472_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_4473_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_4474_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_4475_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_4476_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_4477_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_4478_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_4479_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_4480_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_4481_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_4482_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_4483_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_4484_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_4485_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_4486_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_4487_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_4488_vebt__inst_Oset__vebt__insert,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( vEBT_set_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ) ).

% vebt_inst.set_vebt_insert
tff(fact_4489_VEBT__internal_Oinsert__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) = vEBT_set_vebt(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)) ) ) ) ).

% VEBT_internal.insert_correct
tff(fact_4490_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A7: set(B)] :
      ( ~ aa(set(A),$o,member(A,B2),A3)
     => ( ~ aa(set(B),$o,member(B,aa(A,B,F2,B2)),A7)
       => ( bij_betw(A,B,F2,A3,A7)
         => 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)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A7),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_4491_set__vebt__minNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_VEBT_minNull(T2)
      <=> ( vEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% set_vebt_minNull
tff(fact_4492_set__union,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] : aa(list(A),set(A),set2(A),union(A,Xsa,Ysa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) ).

% set_union
tff(fact_4493_VEBT__internal_Oinsert_H__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T2,X)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat))))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ) ).

% VEBT_internal.insert'_correct
tff(fact_4494_set__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ).

% set_vebt_delete
tff(fact_4495_invar__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => vEBT_invar_vebt(vEBT_vebt_delete(T2,X),n) ) ).

% invar_vebt_delete
tff(fact_4496_IntI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),A3)
     => ( aa(set(A),$o,member(A,C2),B3)
       => aa(set(A),$o,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_4497_Int__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( aa(set(A),$o,member(A,C2),A3)
        & aa(set(A),$o,member(A,C2),B3) ) ) ).

% Int_iff
tff(fact_4498_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).

% le_inf_iff
tff(fact_4499_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_4500_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3) ) ) ).

% Int_subset_iff
tff(fact_4501_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,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)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_insert_right_if1
tff(fact_4502_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,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)),insert(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_4503_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)),insert(A),A2),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% insert_inter_insert
tff(fact_4504_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,A2),C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_4505_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,A2),C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3) ) ) ).

% Int_insert_left_if0
tff(fact_4506_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T5: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T5)) = T5 ).

% Int_Un_eq(4)
tff(fact_4507_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)) = S2 ).

% Int_Un_eq(3)
tff(fact_4508_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)),T5) = T5 ).

% Int_Un_eq(2)
tff(fact_4509_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4510_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T5: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T5)) = T5 ).

% Un_Int_eq(4)
tff(fact_4511_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)) = S2 ).

% Un_Int_eq(3)
tff(fact_4512_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)),T5) = T5 ).

% Un_Int_eq(2)
tff(fact_4513_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T5: 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),T5)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4514_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) )
    <=> ( ~ aa(set(A),$o,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_4515_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B3: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,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_4516_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)),B3) )
    <=> ( ~ aa(set(A),$o,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_4517_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)),B3) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,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_4518_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),minus_minus(set(A),B3),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_4519_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_4520_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_4521_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(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_4522_Un__Int__crazy,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_crazy
tff(fact_4523_Int__Un__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_Un_distrib
tff(fact_4524_Un__Int__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_Int_distrib
tff(fact_4525_Int__Un__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) ).

% Int_Un_distrib2
tff(fact_4526_Un__Int__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_distrib2
tff(fact_4527_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_4528_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_4529_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_4530_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_4531_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_4532_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_4533_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X) ) ) ).

% less_infI2
tff(fact_4534_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X) ) ) ).

% less_infI1
tff(fact_4535_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_4536_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).

% inf_sup_ord(1)
tff(fact_4537_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).

% inf_le1
tff(fact_4538_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).

% inf_le2
tff(fact_4539_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2) ) ) ) ).

% le_infE
tff(fact_4540_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_4541_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D3)) ) ) ) ).

% inf_mono
tff(fact_4542_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X) ) ) ).

% le_infI1
tff(fact_4543_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X) ) ) ).

% le_infI2
tff(fact_4544_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_4545_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_4546_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y3)),X4)
         => ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y3)),Y3)
           => ( ! [X4: A,Y3: A,Z: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,Y3),Z)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_4547_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_4548_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_4549_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_4550_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_4551_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_4552_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_4553_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_4554_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ) ).

% inf_greatest
tff(fact_4555_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_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),C3) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Diff_Int_distrib2
tff(fact_4563_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),aa(set(A),set(A),minus_minus(set(A),A3),B3)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B3)) ).

% Diff_Int_distrib
tff(fact_4564_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(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_4565_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),B3) ).

% Diff_Int2
tff(fact_4566_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),C3)) ).

% Int_Diff
tff(fact_4567_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),B3)
             => ( X2 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4568_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_4569_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_4570_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => ~ aa(set(A),$o,member(A,X2),B3) ) ) ).

% disjoint_iff
tff(fact_4571_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => ~ aa(set(A),$o,member(A,X4),B3) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_4572_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D)
       => aa(set(A),$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)),C3),D)) ) ) ).

% Int_mono
tff(fact_4573_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_4574_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_4575_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_4576_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_4577_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_greatest
tff(fact_4578_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),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A3)
           => ( aa(A,$o,P,X4)
             => aa(A,$o,Q,X4) ) )
       => 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),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(fun(A,$o),set(A),collect(A),Q))) ) ) ).

% Int_Collect_mono
tff(fact_4579_Int__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = $ite(aa(set(A),$o,member(A,A2),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_insert_left
tff(fact_4580_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)),insert(A),A2),B3)) = $ite(aa(set(A),$o,member(A,A2),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),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_4581_Int__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jw(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Int_def
tff(fact_4582_Int__Collect,axiom,
    ! [A: $tType,X: A,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))
    <=> ( aa(set(A),$o,member(A,X),A3)
        & aa(A,$o,P,X) ) ) ).

% Int_Collect
tff(fact_4583_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_4584_IntE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => ~ ( aa(set(A),$o,member(A,C2),A3)
         => ~ aa(set(A),$o,member(A,C2),B3) ) ) ).

% IntE
tff(fact_4585_IntD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => aa(set(A),$o,member(A,C2),A3) ) ).

% IntD1
tff(fact_4586_IntD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => aa(set(A),$o,member(A,C2),B3) ) ).

% IntD2
tff(fact_4587_Int__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_assoc
tff(fact_4588_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_4589_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_4590_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_4591_Int__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_left_commute
tff(fact_4592_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2))) ) ).

% distrib_sup_le
tff(fact_4593_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,$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),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ).

% distrib_inf_le
tff(fact_4594_VEBT__internal_OminNull__delete__time__bound_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T2,X))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T2,X)),one_one(nat)) ) ) ).

% VEBT_internal.minNull_delete_time_bound'
tff(fact_4595_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),minus_minus(set(A),A3),B3)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_4596_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),minus_minus(set(A),A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_4597_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3) ) ).

% Un_Int_assoc_eq
tff(fact_4598_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),aa(set(A),set(A),minus_minus(set(A),A3),C3)) ).

% Diff_Un
tff(fact_4599_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),aa(set(A),set(A),minus_minus(set(A),A3),C3)) ).

% Diff_Int
tff(fact_4600_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),minus_minus(set(A),A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_4601_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),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_4602_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_4603_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_4604_Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(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_4605_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_4606_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q3: A,R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R3))
        <=> aa(A,$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),Q3))),R3) ) ) ).

% sup_neg_inf
tff(fact_4607_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ) ).

% shunt2
tff(fact_4608_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2)) ) ) ).

% shunt1
tff(fact_4609_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_4610_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C3: set(B),G: fun(A,B),B3: set(A),D: set(B)] :
      ( bij_betw(A,B,F2,A3,C3)
     => ( bij_betw(A,B,G,B3,D)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C3),D) = 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_jy(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C3),D)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_4611_VEBT__internal_OminNull__delete__time__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T2,X))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T2,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% VEBT_internal.minNull_delete_time_bound
tff(fact_4612_vebt__inst_Oset__vebt__minNull,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
      <=> ( vEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% vebt_inst.set_vebt_minNull
tff(fact_4613_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,X: 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)),insert(A),X),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_4614_vebt__inst_Oset__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ).

% vebt_inst.set_vebt_delete
tff(fact_4615_insertsimp,axiom,
    ! [T2: vEBT_VEBT,Na: nat,L: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_4616_VEBT__internal_Odelete__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ).

% VEBT_internal.delete_correct
tff(fact_4617_VEBT__internal_Odelete__correct_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),X),bot_bot(set(nat)))) ) ) ).

% VEBT_internal.delete_correct'
tff(fact_4618_set__encode__insert,axiom,
    ! [A3: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( ~ aa(set(nat),$o,member(nat,Na),A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),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))),Na)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_4619_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ig(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_4620_List_Ofinite__set,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xsa)) ).

% List.finite_set
tff(fact_4621_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),F3) = zero_zero(B) )
          <=> ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),F3)
               => ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4622_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_4623_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4624_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_4625_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)
         => ( ( groups7121269368397514597t_prod(A,B,F2,A3) = zero_zero(B) )
          <=> ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
                & ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_4626_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_4627_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Na) ) ) ) ) ).

% ivl_subset
tff(fact_4628_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_4629_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_4630_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_4631_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)
         => ( aa(set(A),$o,member(A,A2),A3)
           => dvd_dvd(B,aa(A,B,F2,A2),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ).

% dvd_prodI
tff(fact_4632_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)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ( ( B2 = aa(A,B,F2,A2) )
             => dvd_dvd(B,B2,groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_4633_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Na)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Na)) = set_or7035219750837199246ssThan(A,Na,M) ) ) ) ).

% ivl_diff
tff(fact_4634_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_jz(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(aa(set(A),$o,member(A,A2),S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_4635_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_ka(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(aa(set(A),$o,member(A,A2),S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_4636_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)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kb(A,fun(fun(A,B),fun(A,B)),A2),B2),S2) = $ite(aa(set(A),$o,member(A,A2),S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_4637_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)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kc(A,fun(fun(A,B),fun(A,B)),A2),B2),S2) = $ite(aa(set(A),$o,member(A,A2),S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_4638_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_kd(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_4639_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_ke(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_4640_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,X),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)),insert(A),X),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_4641_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,X),A3)
           => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_4642_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)),groups7121269368397514597t_prod(A,nat,F2,A3))
      <=> ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X2)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4643_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_4644_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),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_kf(fun(A,$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),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_4645_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_kg(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_4646_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_4647_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.op_ivl_Suc
tff(fact_4648_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_kh(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_4649_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D3: 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_ki(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D3,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_4650_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),B3))) ).

% inf_set_def
tff(fact_4651_inf__Int__eq,axiom,
    ! [A: $tType,R2: set(A),S2: set(A),X3: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),R2)),aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R2),S2)) ) ).

% inf_Int_eq
tff(fact_4652_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_kj(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_4653_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),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_kk(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_4654_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$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_km(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R2)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_kp(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2)),B3) ) ) ) ) ).

% sum.swap_restrict
tff(fact_4655_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4656_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_4657_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_4658_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4659_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4660_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
            <=> ( ( A2 = C2 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4661_bounded__nat__set__is__finite,axiom,
    ! [N3: set(nat),Na: nat] :
      ( ! [X4: nat] :
          ( aa(set(nat),$o,member(nat,X4),N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Na) )
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% bounded_nat_set_is_finite
tff(fact_4662_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M3: nat] :
        ! [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4663_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M3: nat] :
        ! [X2: nat] :
          ( aa(set(nat),$o,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_4664_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( 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_kq(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R2),A3) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ks(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2),B3) ) ) ) ) ).

% prod.swap_restrict
tff(fact_4665_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4666_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_4667_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
         => aa(nat,$o,P,M3) )
    <=> ! [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X2) ) ) ).

% all_nat_less_eq
tff(fact_4668_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
          & aa(nat,$o,P,M3) )
    <=> ? [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X2) ) ) ).

% ex_nat_less_eq
tff(fact_4669_lessThan__atLeast0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Na) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ).

% lessThan_atLeast0
tff(fact_4670_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => ? [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),X5)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa3) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% infinite_growing
tff(fact_4671_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)) )
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),S2)
                & ~ ? [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),S2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X4) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4672_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),I5) )
         => ( ! [I2: B] :
                ( aa(set(B),$o,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)) )
           => ( aa(set(B),$o,member(B,I),I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F2,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4673_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_4674_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)
         => ( ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),A3)
                & ( aa(A,B,F2,X3) = zero_zero(B) ) )
           => ( groups7121269368397514597t_prod(A,B,F2,A3) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_4675_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_4676_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] :
                ( ~ aa(set(nat),$o,member(nat,N),N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_4677_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4678_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: 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),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hx(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4679_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4680_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ku(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_4681_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kv(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kv(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_kw(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_4682_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4683_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),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(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,$o),fun(A,B),aTP_Lamp_kx(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_4684_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_4685_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),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ky(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A3) ) ) ) ).

% prod.inter_filter
tff(fact_4686_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_kz(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_4687_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D3: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D3)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( 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,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4688_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D3: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D3)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A2,B2)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4689_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_4690_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Na,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4691_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_4692_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,P3: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Na,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4693_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,P3))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4694_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),T2: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( ! [X4: B] :
                  ( aa(set(B),$o,member(B,X4),T2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X4)) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S)
                   => ? [Xa3: B] :
                        ( aa(set(B),$o,member(B,Xa3),T2)
                        & ( aa(B,A,I,Xa3) = X4 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X4)),aa(B,C,G,Xa3)) ) )
               => 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),T2)) ) ) ) ) ) ).

% sum_le_included
tff(fact_4695_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)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3) = zero_zero(B) )
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4696_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)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => ( ? [X3: A] :
                  ( aa(set(A),$o,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_ex1
tff(fact_4697_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,$o)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R2,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),$o,finite_finite2(B),S2)
             => ( ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),S2)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X4)),aa(B,A,G,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,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_4698_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)))
           => ( ! [X4: A,S4: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S4)
                 => ( ! [Y2: A] :
                        ( aa(set(A),$o,member(A,Y2),S4)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X4)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),S4)) ) ) )
             => aa(set(A),$o,P,S2) ) ) ) ) ).

% finite_ranking_induct
tff(fact_4699_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,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B4) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4700_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,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4701_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)) )
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$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_4702_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_4703_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$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)),insert(A),X),A3)) = $ite(aa(set(A),$o,member(A,X),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,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_4704_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T6: set(B),S2: set(A),I: fun(B,A),J: fun(A,B),T5: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),S2),S5))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),S2),S5))
                   => aa(set(B),$o,member(B,aa(A,B,J,A4)),aa(set(B),set(B),minus_minus(set(B),T5),T6)) )
               => ( ! [B4: B] :
                      ( aa(set(B),$o,member(B,B4),aa(set(B),set(B),minus_minus(set(B),T5),T6))
                     => ( aa(A,B,J,aa(B,A,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( aa(set(B),$o,member(B,B4),aa(set(B),set(B),minus_minus(set(B),T5),T6))
                       => aa(set(A),$o,member(A,aa(B,A,I,B4)),aa(set(A),set(A),minus_minus(set(A),S2),S5)) )
                   => ( ! [A4: A] :
                          ( aa(set(A),$o,member(A,A4),S5)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( aa(set(B),$o,member(B,B4),T6)
                           => ( aa(B,C,H,B4) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( aa(set(A),$o,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),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4705_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = $ite(aa(set(A),$o,member(A,X),A3),groups7121269368397514597t_prod(A,B,G,A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,A3))) ) ) ) ).

% prod.insert_if
tff(fact_4706_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] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => dvd_dvd(B,aa(A,B,F2,A4),aa(A,B,G,A4)) )
             => dvd_dvd(B,groups7121269368397514597t_prod(A,B,F2,A3),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_4707_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,groups7121269368397514597t_prod(A,B,F2,A3),groups7121269368397514597t_prod(A,B,F2,B3)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_4708_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] :
            ( aa(set(A),$o,member(A,X2),A3)
            & ( aa(A,nat,F2,X2) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( ( X2 != Xa2 )
                 => ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4709_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] :
            ( aa(set(A),$o,member(A,X2),A3)
            & ( aa(A,nat,F2,X2) = one_one(nat) )
            & ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( ( X2 != Xa2 )
                 => ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4710_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = zero_zero(B) )
             => ( aa(set(A),$o,member(A,I),S)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4711_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B3: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = B3 )
             => ( aa(set(A),$o,member(A,I),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B3) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4712_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_4713_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_la(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_4714_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),minus_minus(set(A),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lb(fun(A,B),fun(A,$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_4715_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)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = groups7121269368397514597t_prod(A,B,aa(set(A),fun(A,B),aTP_Lamp_lc(fun(A,B),fun(set(A),fun(A,B)),G),B3),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_4716_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ld(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_4717_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)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_le(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4718_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] :
                ( ~ aa(set(nat),$o,member(nat,N),N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => aa(A,$o,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_4719_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_kd(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(fun(nat,$o),set(nat),collect(nat),P))) ) ) ).

% sums_If_finite
tff(fact_4720_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)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ke(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_4721_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] :
                ( ~ aa(set(nat),$o,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_4722_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lf(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_4723_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4724_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)
         => ( aa(B,B,exp(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_lg(fun(A,B),fun(A,B),F2),I5) ) ) ) ).

% exp_sum
tff(fact_4725_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4726_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4727_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_4728_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_4729_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4730_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4731_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_4732_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G,Na)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4733_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4734_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4735_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)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4736_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% sum.last_plus
tff(fact_4737_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( aa(set(A),$o,member(A,I),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( aa(set(A),$o,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_4738_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Na)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% prod.last_plus
tff(fact_4739_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] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$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_4740_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( aa(set(A),$o,member(A,I),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( aa(set(A),$o,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)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4741_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] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$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)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_4742_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),C3),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),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4743_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),C3),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),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C3) )
                   => ( 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_4744_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = 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),T5) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4745_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4746_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,H,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( 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),T5) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4747_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T5) = 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_4748_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),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_4749_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),minus_minus(set(A),A3),B3)) = aa(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_4750_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),S2),T5))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T5))
                     => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
                 => ( 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),T5) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4751_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)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),B3))),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_4752_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),C3),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B3) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C3) = groups7121269368397514597t_prod(A,B,H,C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4753_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),C3),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C3) = groups7121269368397514597t_prod(A,B,H,C3) )
                   => ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4754_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,S2) = groups7121269368397514597t_prod(A,B,G,T5) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4755_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T5) = groups7121269368397514597t_prod(A,B,G,S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4756_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,H,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( groups7121269368397514597t_prod(A,B,G,S2) = groups7121269368397514597t_prod(A,B,H,T5) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4757_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T5) = groups7121269368397514597t_prod(A,B,H,S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4758_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_4759_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A3,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_4760_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_4761_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hd(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4762_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Na,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_lh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4763_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),minus_minus(set(A),A3),B3)) = aa(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_4764_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_li(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_4765_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_lj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4766_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_lk(fun(nat,fun(nat,A)),fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_4767_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ll(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = 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),Na),K))) ) ).

% sum.nat_group
tff(fact_4768_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lm(fun(nat,A),fun(nat,fun(nat,A)),G),K),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = groups7121269368397514597t_prod(nat,A,G,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))) ) ).

% prod.nat_group
tff(fact_4769_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ln(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_4770_prod__Suc__fact,axiom,
    ! [Na: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_fact
tff(fact_4771_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_4772_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T6: set(B),H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S2),S5),aa(set(B),set(B),minus_minus(set(B),T5),T6))
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),S5)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B4: B] :
                      ( aa(set(B),$o,member(B,B4),T6)
                     => ( 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_hw(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),T5) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4773_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S2: A,A3: set(nat),S5: A,F2: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S2)
         => ( aa(set(nat),$o,finite_finite2(nat),A3)
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_lo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_lp(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2)),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4774_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T6: set(B),H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S2),S5),aa(set(B),set(B),minus_minus(set(B),T5),T6))
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),S5)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B4: B] :
                      ( aa(set(B),$o,member(B,B4),T6)
                     => ( aa(B,C,G,B4) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_is(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S2) = groups7121269368397514597t_prod(B,C,G,T5) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4775_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),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_lq(fun(A,$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),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_4776_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),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_lr(fun(A,$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),groups7121269368397514597t_prod(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_4777_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_4778_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.head_if
tff(fact_4779_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.head_if
tff(fact_4780_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] :
                ( aa(set(A),$o,member(A,I2),A3)
               => ( aa(B,$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),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_4781_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)),groups7121269368397514597t_prod(A,B,F2,A3))
          <=> ? [X2: A] :
                ( aa(set(A),$o,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_4782_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] :
                  ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),B3),A3))
                 => aa(B,$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_4783_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),$o,finite_finite2(A),B3)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,G,X4) = 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_4784_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),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,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_4785_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$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)),insert(A),X),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4786_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),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(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_4787_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_4788_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_4789_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] :
                  ( aa(set(nat),$o,member(nat,N),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$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_4790_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_Suc
tff(fact_4791_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Na,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4792_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),pred_numeral(K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_4793_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4794_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,Na) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jm(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_4795_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)
         => ( aa(B,B,aa(B,fun(B,B),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_ls(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),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_lt(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_lu(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_4796_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_lv(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S2) = $ite(aa(set(A),$o,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),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_4797_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)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_lw(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2),S2) = $ite(aa(set(A),$o,member(A,A2),S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_4798_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,minus_minus(nat,Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_rev
tff(fact_4799_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_4800_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)
           => ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),minus_minus(set(A),B3),A3))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),B3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
                 => 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_4801_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] :
                  ( aa(set(A),$o,member(A,B4),aa(set(A),set(A),minus_minus(set(A),B3),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B4)) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,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),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B3)) ) ) ) ) ) ).

% prod_mono2
tff(fact_4802_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4)) )
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_4803_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),$o,finite_finite2(A),B3)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,F2,X4) != zero_zero(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)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B3))),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_4804_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) )
           => ( groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(B,B,aa(B,fun(B,B),divide_divide(B),groups7121269368397514597t_prod(A,B,F2,A3)),aa(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ) ).

% prod_diff1
tff(fact_4805_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( aa(A,$o,sums(A,F2),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_lx(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),S) ) ) ) ).

% sums_group
tff(fact_4806_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I2)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_ly(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_4807_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ) ) ).

% even_set_encode_iff
tff(fact_4808_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_lz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_4809_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4810_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),C2),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4811_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_lessThan(nat),Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4812_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Na),K),Na)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))) ) ) ) ).

% fact_split
tff(fact_4813_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mb(nat,fun(nat,fun(nat,A)),K),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4814_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mc(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4815_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_md(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4816_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_md(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4817_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jr(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4818_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_4819_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
               => aa(A,$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,$o),ord_less(nat),J2),Na)
                 => aa(A,$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),Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_me(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_4820_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),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A3),B3)) ) ).

% finite_Diff_insert
tff(fact_4821_Chebyshev__sum__upper__nat,axiom,
    ! [Na: 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),Na)
           => 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),Na)
             => 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),Na),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_mf(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_4822_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ju(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4823_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4824_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_mg(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_4825_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_mh(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_4826_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)),insert(A),A2),A3))
    <=> aa(set(A),$o,finite_finite2(A),A3) ) ).

% finite_insert
tff(fact_4827_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_mi(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_4828_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_mj(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_4829_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_mk(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_4830_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_4831_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ml(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_4832_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(set(complex),$o,finite_finite2(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_gk(nat,fun(complex,fun(complex,$o)),Na),C2))) ) ).

% finite_nth_roots
tff(fact_4833_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_4834_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_mm(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_4835_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),R2: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => ? [Xa3: B] :
                  ( aa(set(B),$o,member(B,Xa3),B3)
                  & aa(B,$o,aa(A,fun(B,$o),R2,X4),Xa3) ) )
         => ? [X4: B] :
              ( aa(set(B),$o,member(B,X4),B3)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R2),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4836_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% not_finite_existsD
tff(fact_4837_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa3),X4)
                     => ( X4 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4838_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa3)
                     => ( X4 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4839_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_4840_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),T5) ) ) ).

% infinite_super
tff(fact_4841_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_4842_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)),insert(A),A2),A3)) ) ).

% finite.insertI
tff(fact_4843_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A8)
           => ( ! [B9: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B9),A8)
                 => aa(set(A),$o,P,B9) )
             => aa(set(A),$o,P,A8) ) )
       => aa(set(A),$o,P,A3) ) ) ).

% finite_psubset_induct
tff(fact_4844_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa3)
                     => ( X4 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4845_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa3),X4)
                     => ( X4 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4846_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A8: set(A)] :
              ( ? [A4: A] : A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),A8)
             => ~ aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.cases
tff(fact_4847_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
    <=> ( ( A2 = bot_bot(set(A)) )
        | ? [A9: set(A),A10: A] :
            ( ( A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A10),A9) )
            & aa(set(A),$o,finite_finite2(A),A9) ) ) ) ).

% finite.simps
tff(fact_4848_finite__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X4: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X4),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),F5)) ) ) )
         => aa(set(A),$o,P,F3) ) ) ) ).

% finite_induct
tff(fact_4849_finite__ne__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( ( F3 != bot_bot(set(A)) )
       => ( ! [X4: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))))
         => ( ! [X4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ aa(set(A),$o,member(A,X4),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_4850_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
      ( ! [A8: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A8)
         => aa(set(A),$o,P,A8) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X4: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X4),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),F5)) ) ) )
         => aa(set(A),$o,P,A3) ) ) ) ).

% infinite_finite_induct
tff(fact_4851_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A4),A3)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
                   => ( ~ aa(set(A),$o,member(A,A4),F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4852_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A4),A3)
                 => ( ~ aa(set(A),$o,member(A,A4),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_4853_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),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_4854_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X5,A3)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,X5,A8)
           => ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),A8)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% infinite_coinduct
tff(fact_4855_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,A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( aa(set(A),$o,member(A,A4),A8)
               => ( aa(set(A),$o,P,A8)
                 => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_4856_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)))
       => ( ! [A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B3)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% finite_remove_induct
tff(fact_4857_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),$o,P,B3) )
       => ( ! [A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B3)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% remove_induct
tff(fact_4858_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)))
       => ( ! [T7: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T7),S2)
             => ( aa(set(A),$o,P,T7)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),aa(set(A),set(A),minus_minus(set(A),S2),T7))
                    & aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),T7)) ) ) )
         => aa(set(A),$o,P,S2) ) ) ) ).

% finite_induct_select
tff(fact_4859_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_atMost(nat),K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_4860_finite__nat__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_bounded
tff(fact_4861_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K3)) ) ).

% finite_nat_iff_bounded
tff(fact_4862_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,$o),ord_less_eq(int),M3),aa(int,int,abs_abs(int),N2))
          & aa(set(int),$o,member(int,N2),S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_4863_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,$o),ord_less(int),M3),aa(int,int,abs_abs(int),N2))
          & aa(set(int),$o,member(int,N2),S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4864_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,$o),ord_less(nat),M3),N2)
          & aa(set(nat),$o,member(nat,N2),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_4865_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M4)
         => ? [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N8)
              & aa(set(nat),$o,member(nat,N8),S2) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S2) ) ).

% unbounded_k_infinite
tff(fact_4866_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,$o),ord_less_eq(nat),M3),N2)
          & aa(set(nat),$o,member(nat,N2),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_4867_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F3)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4868_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)),insert(A),the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_4869_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral(int) ).

% Code_Target_Int.positive_def
tff(fact_4870_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_4871_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_4872_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(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)),insert(A),I),I5)) = $ite(aa(set(A),$o,member(A,I),I5),groups1027152243600224163dd_sum(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I5))) ) ) ) ).

% sum.insert'
tff(fact_4873_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mo(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_4874_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X4: A,Y3: A] :
            ( aa(set(A),$o,member(A,X4),A3)
           => ( aa(set(A),$o,member(A,Y3),A3)
             => ( X4 = Y3 ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_4875_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_mp(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_4876_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S2)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T5) = groups1027152243600224163dd_sum(A,B,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4877_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S2)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4878_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T5) = groups1027152243600224163dd_sum(A,B,G,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4879_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,G,T5) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4880_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mp(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_4881_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mo(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mo(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_4882_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)),insert(A),X2),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_4883_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X4: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_4884_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_4885_csqrt_Osimps_I1_J,axiom,
    ! [Z2: complex] : re(csqrt(Z2)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z2)),re(Z2))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% csqrt.simps(1)
tff(fact_4886_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),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(A,$o)),A3),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_4887_insert_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% insert'_bound_height
tff(fact_4888_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Na))) = Na ).

% card_Collect_less_nat
tff(fact_4889_complex__Re__of__nat,axiom,
    ! [Na: nat] : re(aa(nat,complex,semiring_1_of_nat(complex),Na)) = aa(nat,real,semiring_1_of_nat(real),Na) ).

% complex_Re_of_nat
tff(fact_4890_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ml(nat,fun(nat,$o),Na))) = aa(nat,nat,suc,Na) ).

% card_Collect_le_nat
tff(fact_4891_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_4892_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_4893_complex__Re__numeral,axiom,
    ! [V2: num] : re(aa(num,complex,numeral_numeral(complex),V2)) = aa(num,real,numeral_numeral(real),V2) ).

% complex_Re_numeral
tff(fact_4894_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_mr(A,fun(B,A),Y),A3) = aa(nat,A,power_power(A,Y),aa(set(B),nat,finite_card(B),A3)) ) ).

% prod_constant
tff(fact_4895_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_4896_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ aa(set(A),$o,member(A,X),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_4897_Re__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : re(aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),F2),S)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_ms(fun(A,complex),fun(A,real),F2)),S) ).

% Re_sum
tff(fact_4898_Re__divide__of__nat,axiom,
    ! [Z2: complex,Na: nat] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z2),aa(nat,complex,semiring_1_of_nat(complex),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z2)),aa(nat,real,semiring_1_of_nat(real),Na)) ).

% Re_divide_of_nat
tff(fact_4899_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_mt(A,fun(B,A),Y)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant
tff(fact_4900_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( ~ aa(set(A),$o,member(A,A2),B3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_4901_Re__divide__numeral,axiom,
    ! [Z2: complex,W2: num] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z2),aa(num,complex,numeral_numeral(complex),W2))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z2)),aa(num,real,numeral_numeral(real),W2)) ).

% Re_divide_numeral
tff(fact_4902_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero(complex) )
     => ( ( re(Y) = zero_zero(real) )
       => ( cos(real,arg(Y)) = zero_zero(real) ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_4903_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_mu(fun(A,$o),fun(A,B),P)),A3) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_4904_sums__Re,axiom,
    ! [X5: fun(nat,complex),A2: complex] :
      ( aa(complex,$o,sums(complex,X5),A2)
     => aa(real,$o,sums(real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),X5)),re(A2)) ) ).

% sums_Re
tff(fact_4905_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)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_mw(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_4906_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_4907_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
          & ( aa(set(A),nat,finite_card(A),B5) = Na )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_4908_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B3: set(A),A3: set(B),R3: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( ! [A4: B] :
            ( aa(set(B),$o,member(B,A4),A3)
           => ? [B10: A] :
                ( aa(set(A),$o,member(A,B10),B3)
                & aa(A,$o,aa(B,fun(A,$o),R3,A4),B10) ) )
       => ( ! [A1: B,A22: B,B4: A] :
              ( aa(set(B),$o,member(B,A1),A3)
             => ( aa(set(B),$o,member(B,A22),A3)
               => ( aa(set(A),$o,member(A,B4),B3)
                 => ( aa(A,$o,aa(B,fun(A,$o),R3,A1),B4)
                   => ( aa(A,$o,aa(B,fun(A,$o),R3,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => 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_4909_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : aa(nat,$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)),insert(A),X),A3))) ).

% card_insert_le
tff(fact_4910_Cauchy__Re,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Re
tff(fact_4911_imaginary__unit_Osimps_I1_J,axiom,
    re(imaginary_unit) = zero_zero(real) ).

% imaginary_unit.simps(1)
tff(fact_4912_complex__Re__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(X)),real_V7770717601297561774m_norm(complex,X)) ).

% complex_Re_le_cmod
tff(fact_4913_zero__complex_Osimps_I1_J,axiom,
    re(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(1)
tff(fact_4914_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R2: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R2),X4))) = aa(B,nat,K,X4) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_mx(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R2)),S) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_4915_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_my(A,nat)),A3) ).

% card_eq_sum
tff(fact_4916_card__2__iff_H,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X2: A] :
          ( aa(set(A),$o,member(A,X2),S2)
          & ? [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),S2)
              & ( X2 != Xa2 )
              & ! [Xb4: A] :
                  ( aa(set(A),$o,member(A,Xb4),S2)
                 => ( ( Xb4 = X2 )
                    | ( Xb4 = Xa2 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_4917_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_4918_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_4919_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B11: A,B12: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B11),B12) )
          & ~ aa(set(A),$o,member(A,B11),B12)
          & ( aa(set(A),nat,finite_card(A),B12) = K )
          & aa(set(A),$o,finite_finite2(A),B12) ) ) ).

% card_Suc_eq_finite
tff(fact_4920_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = $ite(aa(set(A),$o,member(A,X),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_4921_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Na: nat,S2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),S2))
     => ~ ! [T7: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T7),S2)
           => ( ( aa(set(A),nat,finite_card(A),T7) = Na )
             => ~ aa(set(A),$o,finite_finite2(A),T7) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4922_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F3: set(A),C3: nat] :
      ( ! [G2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G2),F3)
         => ( aa(set(A),$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)),C3) ) )
     => ( aa(set(A),$o,finite_finite2(A),F3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F3)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4923_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_4924_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_4925_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X4: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_4926_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),minus_minus(set(A),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4927_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),minus_minus(set(A),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B3),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4928_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_4929_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M7: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M7)
     => ? [H3: fun(A,nat)] : bij_betw(A,nat,H3,M7,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M7))) ) ).

% ex_bij_betw_finite_nat
tff(fact_4930_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_4931_summable__Re,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Re
tff(fact_4932_abs__Re__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),real_V7770717601297561774m_norm(complex,X)) ).

% abs_Re_le_cmod
tff(fact_4933_Re__csqrt,axiom,
    ! [Z2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(csqrt(Z2))) ).

% Re_csqrt
tff(fact_4934_card__less,axiom,
    ! [M7: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M7)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_mz(set(nat),fun(nat,fun(nat,$o)),M7),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_4935_card__less__Suc,axiom,
    ! [M7: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M7)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_na(set(nat),fun(nat,fun(nat,$o)),M7),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_mz(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc
tff(fact_4936_card__less__Suc2,axiom,
    ! [M7: set(nat),I: nat] :
      ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M7)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_na(set(nat),fun(nat,fun(nat,$o)),M7),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_mz(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc2
tff(fact_4937_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_4938_sum__constant__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_nb(A,fun(B,A),Y)),A3) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant_scaleR
tff(fact_4939_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_nc(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_4940_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_4941_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T5: set(B),R2: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T5)
       => ( ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),T5)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R2),X4))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_mx(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T5),R2)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T5)) ) ) ) ) ).

% sum_multicount
tff(fact_4942_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A3)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_nd(A,real)),A3) ).

% real_of_card
tff(fact_4943_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K6) )
         => 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))),K6)) ) ) ).

% sum_bounded_above
tff(fact_4944_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K6: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K6),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))),K6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% sum_bounded_below
tff(fact_4945_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] :
            ( aa(set(A),$o,member(A,X2),A3)
           => ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( X2 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4946_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)),insert(A),X2),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_4947_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B4: A,B5: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),B5) )
          & ~ aa(set(A),$o,member(A,B4),B5)
          & ( aa(set(A),nat,finite_card(A),B5) = K )
          & ( ( K = zero_zero(nat) )
           => ( B5 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_4948_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B11: A,B12: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B11),B12) )
          & ~ aa(set(A),$o,member(A,B11),B12)
          & ( aa(set(A),nat,finite_card(A),B12) = K )
          & ( ( K = zero_zero(nat) )
           => ( B12 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_4949_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_4950_card__le__Suc__iff,axiom,
    ! [A: $tType,Na: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),A3))
    <=> ? [A10: A,B12: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A10),B12) )
          & ~ aa(set(A),$o,member(A,A10),B12)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),B12))
          & aa(set(A),$o,finite_finite2(A),B12) ) ) ).

% card_le_Suc_iff
tff(fact_4951_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff1_le
tff(fact_4952_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),minus_minus(set(A),A3),B3)) = aa(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_4953_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_4954_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,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),minus_minus(set(A),A3),B3))) ) ).

% diff_card_le_card_Diff
tff(fact_4955_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M7: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M7)
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M7)),M7) ) ).

% ex_bij_betw_nat_finite
tff(fact_4956_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ln(nat,fun(A,$o),Na)))),Na) ) ) ).

% card_roots_unity
tff(fact_4957_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N3)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4958_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_ig(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_ig(nat,nat)),S2)) ).

% card_sum_le_nat_sum
tff(fact_4959_card__nth__roots,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_js(complex,fun(nat,fun(complex,$o)),C2),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_4960_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_go(nat,fun(complex,$o),Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_4961_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,Y5: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),bot_bot(set(A)))) )
          & ( X2 != Y5 ) ) ) ).

% card_2_iff
tff(fact_4962_card__3__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X2: A,Y5: A,Z6: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z6),bot_bot(set(A))))) )
          & ( X2 != Y5 )
          & ( Y5 != Z6 )
          & ( X2 != Z6 ) ) ) ).

% card_3_iff
tff(fact_4963_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_4964_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,X),A3)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_4965_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_4966_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,X),A3)
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_4967_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))
    <=> ( aa(set(A),$o,finite_finite2(A),A3)
        & aa(set(A),$o,member(A,X),A3) ) ) ).

% card_Diff1_less_iff
tff(fact_4968_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),X: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,X),A3)
       => ( aa(set(A),$o,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),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_Diff2_less
tff(fact_4969_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,X),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Diff1_less
tff(fact_4970_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,X),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_4971_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,X),A3),aa(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_4972_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S2: set(A),F2: fun(A,B),K6: real] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),S2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),K6) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),S2))),K6)) ) ) ).

% sum_norm_bound
tff(fact_4973_cmod__plus__Re__le__0__iff,axiom,
    ! [Z2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z2)),re(Z2))),zero_zero(real))
    <=> ( re(Z2) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z2)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_4974_cos__n__Re__cis__pow__n,axiom,
    ! [Na: nat,A2: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A2)) = re(aa(nat,complex,power_power(complex,cis(A2)),Na)) ).

% cos_n_Re_cis_pow_n
tff(fact_4975_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),Na: B,K: nat] :
          ( ! [I2: A] :
              ( aa(set(A),$o,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)),Na) ) )
         => ( 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)),Na)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A3)),aa(nat,B,power_power(B,Na),K)) ) ) ) ) ).

% prod_le_power
tff(fact_4976_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K6) )
         => ( 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))),K6)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_4977_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K6),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)),K6) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4978_card__insert__le__m1,axiom,
    ! [A: $tType,Na: nat,Y: set(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_4979_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)
         => ( groups7121269368397514597t_prod(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_ne(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2),S2) = $ite(aa(set(A),$o,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,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_4980_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ).

% polyfun_roots_card
tff(fact_4981_insertsimp_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,L: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_4982_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4983_csqrt_Ocode,axiom,
    ! [Z2: complex] :
      csqrt(Z2) = complex2(aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z2)),re(Z2))),aa(num,real,numeral_numeral(real),bit0(one2)))),
        aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z2) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z2)))),
          aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z2)),re(Z2))),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% csqrt.code
tff(fact_4984_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] :
      im(csqrt(Z2)) = aa(real,real,
        aa(real,fun(real,real),times_times(real),
          $ite(im(Z2) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z2)))),
        aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z2)),re(Z2))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt.simps(2)
tff(fact_4985_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(X)),zero_zero(real))
       => ( csqrt(X) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(real,real,abs_abs(real),re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_4986_complex__Im__fact,axiom,
    ! [Na: nat] : im(semiring_char_0_fact(complex,Na)) = zero_zero(real) ).

% complex_Im_fact
tff(fact_4987_complex__Im__of__int,axiom,
    ! [Z2: int] : im(aa(int,complex,ring_1_of_int(complex),Z2)) = zero_zero(real) ).

% complex_Im_of_int
tff(fact_4988_Im__complex__of__real,axiom,
    ! [Z2: real] : im(aa(real,complex,real_Vector_of_real(complex),Z2)) = zero_zero(real) ).

% Im_complex_of_real
tff(fact_4989_Im__power__real,axiom,
    ! [X: complex,Na: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( im(aa(nat,complex,power_power(complex,X),Na)) = zero_zero(real) ) ) ).

% Im_power_real
tff(fact_4990_complex__Im__numeral,axiom,
    ! [V2: num] : im(aa(num,complex,numeral_numeral(complex),V2)) = zero_zero(real) ).

% complex_Im_numeral
tff(fact_4991_complex__Im__of__nat,axiom,
    ! [Na: nat] : im(aa(nat,complex,semiring_1_of_nat(complex),Na)) = zero_zero(real) ).

% complex_Im_of_nat
tff(fact_4992_Im__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : im(aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),F2),S)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_nf(fun(A,complex),fun(A,real),F2)),S) ).

% Im_sum
tff(fact_4993_Re__power__real,axiom,
    ! [X: complex,Na: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( re(aa(nat,complex,power_power(complex,X),Na)) = aa(nat,real,power_power(real,re(X)),Na) ) ) ).

% Re_power_real
tff(fact_4994_Im__divide__numeral,axiom,
    ! [Z2: complex,W2: num] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z2),aa(num,complex,numeral_numeral(complex),W2))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z2)),aa(num,real,numeral_numeral(real),W2)) ).

% Im_divide_numeral
tff(fact_4995_Im__divide__of__nat,axiom,
    ! [Z2: complex,Na: nat] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z2),aa(nat,complex,semiring_1_of_nat(complex),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z2)),aa(nat,real,semiring_1_of_nat(real),Na)) ).

% Im_divide_of_nat
tff(fact_4996_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(X))
       => ( csqrt(X) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_4997_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(X)),zero_zero(real))
        | ( ( im(X) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(X)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),X)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_4998_sums__Im,axiom,
    ! [X5: fun(nat,complex),A2: complex] :
      ( aa(complex,$o,sums(complex,X5),A2)
     => aa(real,$o,sums(real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),X5)),im(A2)) ) ).

% sums_Im
tff(fact_4999_Cauchy__Im,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Im
tff(fact_5000_zero__complex_Osimps_I2_J,axiom,
    im(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(2)
tff(fact_5001_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_5002_sums__complex__iff,axiom,
    ! [F2: fun(nat,complex),X: complex] :
      ( aa(complex,$o,sums(complex,F2),X)
    <=> ( aa(real,$o,sums(real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),F2)),re(X))
        & aa(real,$o,sums(real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),F2)),im(X)) ) ) ).

% sums_complex_iff
tff(fact_5003_summable__Im,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Im
tff(fact_5004_complex__is__Int__iff,axiom,
    ! [Z2: complex] :
      ( aa(set(complex),$o,member(complex,Z2),ring_1_Ints(complex))
    <=> ( ( im(Z2) = zero_zero(real) )
        & ? [I4: int] : re(Z2) = aa(int,real,ring_1_of_int(real),I4) ) ) ).

% complex_is_Int_iff
tff(fact_5005_abs__Im__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),real_V7770717601297561774m_norm(complex,X)) ).

% abs_Im_le_cmod
tff(fact_5006_summable__complex__iff,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
    <=> ( summable(real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),F2))
        & summable(real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),F2)) ) ) ).

% summable_complex_iff
tff(fact_5007_Im__eq__0,axiom,
    ! [Z2: complex] :
      ( ( aa(real,real,abs_abs(real),re(Z2)) = real_V7770717601297561774m_norm(complex,Z2) )
     => ( im(Z2) = zero_zero(real) ) ) ).

% Im_eq_0
tff(fact_5008_cmod__eq__Im,axiom,
    ! [Z2: complex] :
      ( ( re(Z2) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z2) = aa(real,real,abs_abs(real),im(Z2)) ) ) ).

% cmod_eq_Im
tff(fact_5009_cmod__eq__Re,axiom,
    ! [Z2: complex] :
      ( ( im(Z2) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z2) = aa(real,real,abs_abs(real),re(Z2)) ) ) ).

% cmod_eq_Re
tff(fact_5010_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( re(X) = re(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),aa(real,real,abs_abs(real),im(Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_5011_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( im(X) = im(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),aa(real,real,abs_abs(real),re(Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_5012_csqrt__principal,axiom,
    ! [Z2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(csqrt(Z2)))
      | ( ( re(csqrt(Z2)) = zero_zero(real) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(csqrt(Z2))) ) ) ).

% csqrt_principal
tff(fact_5013_cmod__le,axiom,
    ! [Z2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z2))),aa(real,real,abs_abs(real),im(Z2)))) ).

% cmod_le
tff(fact_5014_sin__n__Im__cis__pow__n,axiom,
    ! [Na: nat,A2: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A2)) = im(aa(nat,complex,power_power(complex,cis(A2)),Na)) ).

% sin_n_Im_cis_pow_n
tff(fact_5015_cmod__power2,axiom,
    ! [Z2: complex] : aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% cmod_power2
tff(fact_5016_Im__power2,axiom,
    ! [X: complex] : im(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(X))),im(X)) ).

% Im_power2
tff(fact_5017_Re__power2,axiom,
    ! [X: complex] : re(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,minus_minus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Re_power2
tff(fact_5018_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_5019_norm__complex__def,axiom,
    ! [Z2: complex] : real_V7770717601297561774m_norm(complex,Z2) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% norm_complex_def
tff(fact_5020_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(1)
tff(fact_5021_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero(complex) )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% complex_neq_0
tff(fact_5022_Re__divide,axiom,
    ! [X: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Re_divide
tff(fact_5023_csqrt__unique,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( aa(nat,complex,power_power(complex,W2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z2 )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(W2))
          | ( ( re(W2) = zero_zero(real) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(W2)) ) )
       => ( csqrt(Z2) = W2 ) ) ) ).

% csqrt_unique
tff(fact_5024_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(B2))
        | ( ( re(B2) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(B2)) ) )
     => ( csqrt(aa(nat,complex,power_power(complex,B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = B2 ) ) ).

% csqrt_square
tff(fact_5025_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(2)
tff(fact_5026_Im__divide,axiom,
    ! [X: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Im_divide
tff(fact_5027_complex__abs__le__norm,axiom,
    ! [Z2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z2))),aa(real,real,abs_abs(real),im(Z2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),real_V7770717601297561774m_norm(complex,Z2))) ).

% complex_abs_le_norm
tff(fact_5028_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z2)),real_V7770717601297561774m_norm(complex,Z2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z2)),real_V7770717601297561774m_norm(complex,Z2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_5029_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa(complex,complex,inverse_inverse(complex),X) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% inverse_complex.code
tff(fact_5030_Complex__divide,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% Complex_divide
tff(fact_5031_Im__Reals__divide,axiom,
    ! [R3: complex,Z2: complex] :
      ( aa(set(complex),$o,member(complex,R3),real_Vector_Reals(complex))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R3),Z2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R3))),im(Z2))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Im_Reals_divide
tff(fact_5032_Re__Reals__divide,axiom,
    ! [R3: complex,Z2: complex] :
      ( aa(set(complex),$o,member(complex,R3),real_Vector_Reals(complex))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R3),Z2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R3)),re(Z2))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Re_Reals_divide
tff(fact_5033_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N3: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : aa(set(complex),$o,member(complex,aa(nat,complex,G,N)),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [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))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_5034_Reals__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Na: nat] : aa(set(A),$o,member(A,aa(nat,A,semiring_1_of_nat(A),Na)),real_Vector_Reals(A)) ) ).

% Reals_of_nat
tff(fact_5035_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => aa(set(A),$o,member(A,zero_zero(A)),real_Vector_Reals(A)) ) ).

% Reals_0
tff(fact_5036_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,Na: nat] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => aa(set(A),$o,member(A,aa(nat,A,power_power(A,A2),Na)),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_5037_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W2: num] : aa(set(A),$o,member(A,aa(num,A,numeral_numeral(A),W2)),real_Vector_Reals(A)) ) ).

% Reals_numeral
tff(fact_5038_complex__is__Real__iff,axiom,
    ! [Z2: complex] :
      ( aa(set(complex),$o,member(complex,Z2),real_Vector_Reals(complex))
    <=> ( im(Z2) = zero_zero(real) ) ) ).

% complex_is_Real_iff
tff(fact_5039_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_5040_Complex__in__Reals,axiom,
    ! [X: real] : aa(set(complex),$o,member(complex,complex2(X,zero_zero(real))),real_Vector_Reals(complex)) ).

% Complex_in_Reals
tff(fact_5041_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( ( A2 != zero_zero(A) )
           => aa(set(A),$o,member(A,aa(A,A,inverse_inverse(A),A2)),real_Vector_Reals(A)) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_5042_Re__prod__Reals,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,complex)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(complex),$o,member(complex,aa(A,complex,F2,X4)),real_Vector_Reals(complex)) )
     => ( re(groups7121269368397514597t_prod(A,complex,F2,A3)) = groups7121269368397514597t_prod(A,real,aTP_Lamp_ms(fun(A,complex),fun(A,real),F2),A3) ) ) ).

% Re_prod_Reals
tff(fact_5043_complex__mult__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_mult_cnj
tff(fact_5044_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nh(set(A),fun(nat,fun(list(A),$o)),A3),Na))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(set(A),nat,finite_card(A),A3))),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ) ).

% card_lists_length_le
tff(fact_5045_cnj__add__mult__eq__Re,axiom,
    ! [Z2: complex,W2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(W2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z2)),W2)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(W2))))) ).

% cnj_add_mult_eq_Re
tff(fact_5046_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F2),Xsa)) = aa(list(B),nat,size_size(list(B)),Xsa) ).

% length_map
tff(fact_5047_complex__cnj__numeral,axiom,
    ! [W2: num] : cnj(aa(num,complex,numeral_numeral(complex),W2)) = aa(num,complex,numeral_numeral(complex),W2) ).

% complex_cnj_numeral
tff(fact_5048_complex__cnj__of__nat,axiom,
    ! [Na: nat] : cnj(aa(nat,complex,semiring_1_of_nat(complex),Na)) = aa(nat,complex,semiring_1_of_nat(complex),Na) ).

% complex_cnj_of_nat
tff(fact_5049_cnj__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : cnj(aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),F2),S)) = aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),aTP_Lamp_ni(fun(A,complex),fun(A,complex),F2)),S) ).

% cnj_sum
tff(fact_5050_cnj__prod,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : cnj(groups7121269368397514597t_prod(A,complex,F2,S)) = groups7121269368397514597t_prod(A,complex,aTP_Lamp_ni(fun(A,complex),fun(A,complex),F2),S) ).

% cnj_prod
tff(fact_5051_complex__cnj__neg__numeral,axiom,
    ! [W2: num] : cnj(aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W2))) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W2)) ).

% complex_cnj_neg_numeral
tff(fact_5052_complex__In__mult__cnj__zero,axiom,
    ! [Z2: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2))) = zero_zero(real) ).

% complex_In_mult_cnj_zero
tff(fact_5053_finite__maxlen,axiom,
    ! [A: $tType,M7: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M7)
     => ? [N: nat] :
        ! [X3: list(A)] :
          ( aa(set(list(A)),$o,member(list(A),X3),M7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N) ) ) ).

% finite_maxlen
tff(fact_5054_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xsa: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P,Ys) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xsa) ) ).

% length_induct
tff(fact_5055_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_5056_Ex__list__of__length,axiom,
    ! [A: $tType,Na: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = Na ).

% Ex_list_of_length
tff(fact_5057_neq__if__length__neq,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) != aa(list(A),nat,size_size(list(A)),Ysa) )
     => ( Xsa != Ysa ) ) ).

% neq_if_length_neq
tff(fact_5058_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xsa: list(B),G: fun(C,A),Ysa: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xsa) = aa(list(C),list(A),map(C,A,G),Ysa) )
     => ( aa(list(B),nat,size_size(list(B)),Xsa) = aa(list(C),nat,size_size(list(C)),Ysa) ) ) ).

% map_eq_imp_length_eq
tff(fact_5059_sums__cnj,axiom,
    ! [F2: fun(nat,complex),L: complex] :
      ( aa(complex,$o,sums(complex,aTP_Lamp_nj(fun(nat,complex),fun(nat,complex),F2)),cnj(L))
    <=> aa(complex,$o,sums(complex,F2),L) ) ).

% sums_cnj
tff(fact_5060_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_pos_if_in_set
tff(fact_5061_card__length,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% card_length
tff(fact_5062_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nk(set(A),fun(nat,fun(list(A),$o)),A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5063_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_5064_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_5065_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nh(set(A),fun(nat,fun(list(A),$o)),A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5066_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_5067_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_5068_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_5069_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_5070_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_5071_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_5072_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_5073_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_5074_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nk(set(A),fun(nat,fun(list(A),$o)),A3),Na))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A3)),Na) ) ) ).

% card_lists_length_eq
tff(fact_5075_complex__mod__mult__cnj,axiom,
    ! [Z2: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2))) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% complex_mod_mult_cnj
tff(fact_5076_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) )
      & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ) ).

% complex_div_gt_0
tff(fact_5077_complex__norm__square,axiom,
    ! [Z2: complex] : aa(real,complex,real_Vector_of_real(complex),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2)) ).

% complex_norm_square
tff(fact_5078_complex__add__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z2),cnj(Z2)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(Z2))) ).

% complex_add_cnj
tff(fact_5079_VEBT__internal_Oset__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X4,Na) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na) ) ) ).

% VEBT_internal.set_n_deg_not_0
tff(fact_5080_complex__diff__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,minus_minus(complex,Z2),cnj(Z2)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),im(Z2)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_5081_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))),aa(real,complex,real_Vector_of_real(complex),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,B2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_div_cnj
tff(fact_5082_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)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nl(set(A),fun(nat,fun(list(A),$o)),A3),K))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ig(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_5083_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_nm(nat,fun(set(A),fun(list(A),$o)),K),A3))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ig(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_5084_sum__count__set,axiom,
    ! [A: $tType,Xsa: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xsa)),X5) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ) ).

% sum_count_set
tff(fact_5085_distinct__union,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( distinct(A,union(A,Xsa,Ysa))
    <=> distinct(A,Ysa) ) ).

% distinct_union
tff(fact_5086_count__notin,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( aa(A,nat,count_list(A,Xsa),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5087_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_nl(set(A),fun(nat,fun(list(A),$o)),A3),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_5088_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( distinct(A,Xsa)
         => distinct(A,Xsa) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_5089_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_5090_distinct__card,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( distinct(A,Xsa)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ).

% distinct_card
tff(fact_5091_card__distinct,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) )
     => distinct(A,Xsa) ) ).

% card_distinct
tff(fact_5092_count__le__length,axiom,
    ! [A: $tType,Xsa: list(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xsa),X)),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% count_le_length
tff(fact_5093_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_5094_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_5095_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_5096_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_5097_length__subseqs,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xsa)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_subseqs
tff(fact_5098_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: 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),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xsa,Ysa)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ysa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% card_disjoint_shuffles
tff(fact_5099_finite__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),shuffles(A,Xsa,Ysa)) ).

% finite_shuffles
tff(fact_5100_subseqs__refl,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(set(list(A)),$o,member(list(A),Xsa),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa))) ).

% subseqs_refl
tff(fact_5101_shuffles__commutes,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] : shuffles(A,Xsa,Ysa) = shuffles(A,Ysa,Xsa) ).

% shuffles_commutes
tff(fact_5102_subseqs__distinctD,axiom,
    ! [A: $tType,Ysa: list(A),Xsa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Ysa),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa)))
     => ( distinct(A,Xsa)
       => distinct(A,Ysa) ) ) ).

% subseqs_distinctD
tff(fact_5103_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ysa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
     => ( 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)),Xsa)),aa(list(A),nat,size_size(list(A)),Ysa)) ) ) ).

% length_shuffles
tff(fact_5104_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ysa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
     => ( 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),Xsa)),aa(list(A),set(A),set2(A),Ysa)) ) ) ).

% set_shuffles
tff(fact_5105_distinct__set__subseqs,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( distinct(A,Xsa)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xsa))) ) ).

% distinct_set_subseqs
tff(fact_5106_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),Zs: list(A)] :
      ( distinct(A,Xsa)
     => ( distinct(A,Ysa)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
         => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_5107_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Na: 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)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_5108_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : groups4207007520872428315er_sum($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),Na))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_5109_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_5110_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_5111_Pow__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),Y4: set(A)] :
      ( ( pow2(A,X5) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Y4),bot_bot(set(set(A)))) )
    <=> ( ( X5 = bot_bot(set(A)) )
        & ( Y4 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_5112_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(set(A)),$o,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_5113_PowI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(set(A)),$o,member(set(A),A3),pow2(A,B3)) ) ).

% PowI
tff(fact_5114_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,minus_minus(nat,J),I) ).

% length_upt
tff(fact_5115_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_5116_nth__map,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xsa: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F2),Xsa)),Na) = aa(A,B,F2,aa(nat,A,nth(A,Xsa),Na)) ) ) ).

% nth_map
tff(fact_5117_map__nth,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xsa)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) = Xsa ).

% map_nth
tff(fact_5118_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_5119_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_mi(set(A),fun(set(A),$o),A3)) ).

% Pow_def
tff(fact_5120_Pow__top,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),$o,member(set(A),A3),pow2(A,A3)) ).

% Pow_top
tff(fact_5121_PowD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(set(A)),$o,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_5122_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_5123_Pow__bottom,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),pow2(A,B3)) ).

% Pow_bottom
tff(fact_5124_Pow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_5125_distinct__upt,axiom,
    ! [I: nat,J: nat] : distinct(nat,upt(I,J)) ).

% distinct_upt
tff(fact_5126_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Na: nat,M: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,Na),M))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F2),upt(M,Na))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_5127_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( Xsa = Ysa )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ysa) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I4) = aa(nat,A,nth(A,Ysa),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5128_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_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_12) )
    <=> ? [Xs: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs) = 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,Xs),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_5129_nth__equalityI,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ysa) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I2) = aa(nat,A,nth(A,Ysa),I2) ) )
       => ( Xsa = Ysa ) ) ) ).

% nth_equalityI
tff(fact_5130_map__upt__eqI,axiom,
    ! [A: $tType,Xsa: list(A),Na: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,minus_minus(nat,Na),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F2),upt(M,Na)) = Xsa ) ) ) ).

% map_upt_eqI
tff(fact_5131_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_5132_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(M,Na)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na)) ).

% map_Suc_upt
tff(fact_5133_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$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)),Xsa))
         => aa(A,$o,P,aa(nat,A,nth(A,Xsa),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_5134_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o),X: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,P,aa(nat,A,nth(A,Xsa),I2)) )
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
       => aa(A,$o,P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_5135_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xsa))
          & ( aa(nat,A,nth(A,Xsa),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5136_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xsa),Na)) ) ) ).

% list_ball_nth
tff(fact_5137_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
     => aa(set(A),$o,member(A,aa(nat,A,nth(A,Xsa),Na)),aa(list(A),set(A),set2(A),Xsa)) ) ).

% nth_mem
tff(fact_5138_VEBT__internal_Ointhall,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o),Na: nat] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X4) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
       => aa(A,$o,P,aa(nat,A,nth(A,Xsa),Na)) ) ) ).

% VEBT_internal.inthall
tff(fact_5139_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,J: nat] :
      ( distinct(A,Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
         => ( ( aa(nat,A,nth(A,Xsa),I) = aa(nat,A,nth(A,Xsa),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5140_distinct__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( distinct(A,Xsa)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xsa))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xsa))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xsa),I4) != aa(nat,A,nth(A,Xsa),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5141_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or1337092689740270186AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(Na,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_5142_atLeast__upt,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Na)) ).

% atLeast_upt
tff(fact_5143_distinct__Ex1,axiom,
    ! [A: $tType,Xsa: list(A),X: A] :
      ( distinct(A,Xsa)
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
       => ? [X4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xsa))
            & ( aa(nat,A,nth(A,Xsa),X4) = X )
            & ! [Y2: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y2),aa(list(A),nat,size_size(list(A)),Xsa))
                  & ( aa(nat,A,nth(A,Xsa),Y2) = X ) )
               => ( Y2 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5144_map__add__upt,axiom,
    ! [Na: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_nn(nat,fun(nat,nat),Na)),upt(zero_zero(nat),M)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% map_add_upt
tff(fact_5145_atMost__upto,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Na))) ).

% atMost_upto
tff(fact_5146_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_no(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_5147_binomial__def,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,binomial(Na),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_np(nat,fun(nat,fun(set(nat),$o)),Na),K))) ).

% binomial_def
tff(fact_5148_bij__betw__nth,axiom,
    ! [A: $tType,Xsa: list(A),A3: set(nat),B3: set(A)] :
      ( distinct(A,Xsa)
     => ( ( A3 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xsa)) )
       => ( ( B3 = aa(list(A),set(A),set2(A),Xsa) )
         => bij_betw(nat,A,nth(A,Xsa),A3,B3) ) ) ) ).

% bij_betw_nth
tff(fact_5149_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xsa: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xsa) = 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_nq(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xsa))) ) ).

% horner_sum_eq_sum
tff(fact_5150_set__n__lists,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xsa)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_nr(nat,fun(list(A),fun(list(A),$o)),Na),Xsa)) ).

% set_n_lists
tff(fact_5151_set__update__distinct,axiom,
    ! [A: $tType,Xsa: list(A),Na: nat,X: A] :
      ( distinct(A,Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xsa,Na,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xsa),Na)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_5152_list__update__overwrite,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,X: A,Y: A] : list_update(A,list_update(A,Xsa,I,X),I,Y) = list_update(A,Xsa,I,Y) ).

% list_update_overwrite
tff(fact_5153_length__list__update,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,X: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xsa,I,X)) = aa(list(A),nat,size_size(list(A)),Xsa) ).

% length_list_update
tff(fact_5154_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xsa: list(A),X: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,list_update(A,Xsa,I,X)),J) = aa(nat,A,nth(A,Xsa),J) ) ) ).

% nth_list_update_neq
tff(fact_5155_list__update__id,axiom,
    ! [A: $tType,Xsa: list(A),I: nat] : list_update(A,Xsa,I,aa(nat,A,nth(A,Xsa),I)) = Xsa ).

% list_update_id
tff(fact_5156_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_5157_list__update__beyond,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),I)
     => ( list_update(A,Xsa,I,X) = Xsa ) ) ).

% list_update_beyond
tff(fact_5158_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xsa: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,list_update(A,Xsa,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_5159_set__swap,axiom,
    ! [A: $tType,I: nat,Xsa: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xsa,I,aa(nat,A,nth(A,Xsa),J)),J,aa(nat,A,nth(A,Xsa),I))) = aa(list(A),set(A),set2(A),Xsa) ) ) ) ).

% set_swap
tff(fact_5160_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xsa: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( distinct(A,list_update(A,list_update(A,Xsa,I,aa(nat,A,nth(A,Xsa),J)),J,aa(nat,A,nth(A,Xsa),I)))
        <=> distinct(A,Xsa) ) ) ) ).

% distinct_swap
tff(fact_5161_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),K: nat,Y: B] : aa(list(B),list(A),map(B,A,F2),list_update(B,Xsa,K,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F2),Xsa),K,aa(B,A,F2,Y)) ).

% map_update
tff(fact_5162_list__update__swap,axiom,
    ! [A: $tType,I: nat,I6: nat,Xsa: list(A),X: A,X6: A] :
      ( ( I != I6 )
     => ( list_update(A,list_update(A,Xsa,I,X),I6,X6) = list_update(A,list_update(A,Xsa,I6,X6),I,X) ) ) ).

% list_update_swap
tff(fact_5163_set__update__subsetI,axiom,
    ! [A: $tType,Xsa: list(A),A3: set(A),X: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),A3)
     => ( aa(set(A),$o,member(A,X),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,Xsa,I,X))),A3) ) ) ).

% set_update_subsetI
tff(fact_5164_set__update__subset__insert,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,X: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xsa,I,X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xsa))) ).

% set_update_subset_insert
tff(fact_5165_set__update__memI,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
     => aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),list_update(A,Xsa,Na,X))) ) ).

% set_update_memI
tff(fact_5166_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xsa: list(A),X: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,list_update(A,Xsa,I,X)),J) = $ite(I = J,X,aa(nat,A,nth(A,Xsa),J)) ) ) ).

% nth_list_update
tff(fact_5167_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xsa: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( ( list_update(A,Xsa,I,X) = Xsa )
      <=> ( aa(nat,A,nth(A,Xsa),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_5168_length__n__lists,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Na,Xsa)) = aa(nat,nat,power_power(nat,aa(list(A),nat,size_size(list(A)),Xsa)),Na) ).

% length_n_lists
tff(fact_5169_length__n__lists__elem,axiom,
    ! [A: $tType,Ysa: list(A),Na: nat,Xsa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Ysa),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xsa)))
     => ( aa(list(A),nat,size_size(list(A)),Ysa) = Na ) ) ).

% length_n_lists_elem
tff(fact_5170_distinct__list__update,axiom,
    ! [A: $tType,Xsa: list(A),A2: A,I: nat] :
      ( distinct(A,Xsa)
     => ( ~ aa(set(A),$o,member(A,A2),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xsa),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xsa,I,A2)) ) ) ).

% distinct_list_update
tff(fact_5171_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xsa: list(A),X5: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xsa)) = 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_ns(list(A),fun(fun(A,nat),fun(A,nat)),Xsa),F2)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_5172_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,$o)),R22: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R1),R22)
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),accp(A,R22)),accp(A,R1)) ) ).

% accp_subset
tff(fact_5173_accp__subset__induct,axiom,
    ! [A: $tType,D: fun(A,$o),R2: fun(A,fun(A,$o)),X: A,P: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),D),accp(A,R2))
     => ( ! [X4: A,Z: A] :
            ( aa(A,$o,D,X4)
           => ( aa(A,$o,aa(A,fun(A,$o),R2,Z),X4)
             => aa(A,$o,D,Z) ) )
       => ( aa(A,$o,D,X)
         => ( ! [X4: A] :
                ( aa(A,$o,D,X4)
               => ( ! [Z3: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),R2,Z3),X4)
                     => aa(A,$o,P,Z3) )
                 => aa(A,$o,P,X4) ) )
           => aa(A,$o,P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_5174_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X4: A,Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X4),Y3)
         => aa(B,$o,aa(A,fun(B,$o),Q,X4),Y3) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_5175_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Ns))
             => ( X2 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_5176_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_nt(B,A)),Xsa)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_5177_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( groups8242544230860333062m_list(nat,upt(M,Na)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ig(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) ) ) ).

% sum_list_upt
tff(fact_5178_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% predicate2D
tff(fact_5179_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% rev_predicate2D
tff(fact_5180_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xsa: list(A)] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),groups8242544230860333062m_list(A,Xsa)) ) ) ).

% member_le_sum_list
tff(fact_5181_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gs(A,fun(fun(B,A),fun(B,A)),C2),F2)),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xsa))) ) ).

% sum_list_const_mult
tff(fact_5182_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),C2: A,Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_gr(fun(B,A),fun(A,fun(B,A)),F2),C2)),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xsa))),C2) ) ).

% sum_list_mult_const
tff(fact_5183_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gt(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xsa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xsa))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xsa))) ) ).

% sum_list_addf
tff(fact_5184_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gu(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xsa)) = aa(A,A,minus_minus(A,groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xsa))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xsa))) ) ).

% sum_list_subtractf
tff(fact_5185_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xsa)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_5186_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4) )
         => ( ( groups8242544230860333062m_list(A,Xsa) = zero_zero(A) )
          <=> ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
               => ( X2 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_5187_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xsa)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_5188_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xsa: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xsa))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xsa))) ) ).

% sum_list_abs
tff(fact_5189_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xsa))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xsa))) ) ) ).

% sum_list_mono
tff(fact_5190_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xsa: list(A)] :
          ( distinct(A,Xsa)
         => ( groups8242544230860333062m_list(A,Xsa) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_nu(A,A)),aa(list(A),set(A),set2(A),Xsa)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_5191_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_5192_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xsa: list(A),F2: fun(A,B)] :
          ( distinct(A,Xsa)
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xsa)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(list(A),set(A),set2(A),Xsa)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_5193_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xsa: list(A),G: fun(A,B)] :
          ( distinct(A,Xsa)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(list(A),set(A),set2(A),Xsa)) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xsa)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_5194_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_nc(fun(A,nat),fun(A,nat),F2)),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% sum_list_Suc
tff(fact_5195_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R3: A,Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_mt(A,fun(B,A),R3)),Xsa)) = 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)),Xsa))),R3) ) ).

% sum_list_triv
tff(fact_5196_sum__set__upt__conv__sum__list__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(M,Na))) = groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F2),upt(M,Na))) ) ).

% sum_set_upt_conv_sum_list_nat
tff(fact_5197_interv__sum__list__conv__sum__set__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] : groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F2),upt(M,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(M,Na))) ) ).

% interv_sum_list_conv_sum_set_nat
tff(fact_5198_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_nv(nat,fun(nat,fun(list(nat),$o)),M),N3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_nw(nat,fun(nat,fun(list(nat),$o)),M),N3)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_nx(nat,fun(nat,fun(list(nat),$o)),M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5199_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xsa: list(A)] : groups8242544230860333062m_list(A,Xsa) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) ) ).

% sum_list_sum_nth
tff(fact_5200_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_nv(nat,fun(nat,fun(list(nat),$o)),M),N3))) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M)),one_one(nat))),N3) ).

% card_length_sum_list
tff(fact_5201_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xsa)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_ny(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xsa)),aa(list(A),set(A),set2(A),Xsa)) ).

% sum_list_map_eq_sum_count
tff(fact_5202_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xsa: list(A),X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xsa))
         => ( groups8242544230860333062m_list(A,list_update(A,Xsa,K,X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xsa)),X)),aa(nat,A,nth(A,Xsa),K)) ) ) ) ).

% sum_list_update
tff(fact_5203_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( aa(set(int),$o,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ 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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_5204_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa(set(int),$o,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & aa(set(int),$o,member(int,Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
                      & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
                        & ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_5205_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X4: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X4)),aa(nat,A,F2,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$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_5206_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),Xsa: list(B)] :
      ( sorted_wrt(A,R2,aa(list(B),list(A),map(B,A,F2),Xsa))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_nz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F2),Xsa) ) ).

% sorted_wrt_map
tff(fact_5207_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),Xsa) ) ) ).

% strict_sorted_imp_sorted
tff(fact_5208_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),S2))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).

% pred_subset_eq2
tff(fact_5209_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Y3: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),R3)
         => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),S) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R3),S) ) ).

% subrelI
tff(fact_5210_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X4: A,Y3: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => ( aa(set(A),$o,member(A,Y3),aa(list(A),set(A),set2(A),Xsa))
           => ( aa(A,$o,aa(A,fun(A,$o),P,X4),Y3)
             => aa(A,$o,aa(A,fun(A,$o),Q,X4),Y3) ) ) )
     => ( sorted_wrt(A,P,Xsa)
       => sorted_wrt(A,Q,Xsa) ) ) ).

% sorted_wrt_mono_rel
tff(fact_5211_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ysa: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xsa)
         => ( sorted_wrt(A,ord_less(A),Ysa)
           => ( ( aa(list(A),set(A),set2(A),Ysa) = aa(list(A),set(A),set2(A),Xsa) )
             => ( Ysa = Xsa ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_5212_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X2: A,Xa2: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2)),R2)
        <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2)),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_5213_sorted__wrt__true,axiom,
    ! [A: $tType,Xsa: list(A)] : sorted_wrt(A,aTP_Lamp_ob(A,fun(A,$o)),Xsa) ).

% sorted_wrt_true
tff(fact_5214_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R3: A,S: B,R2: set(product_prod(A,B)),S6: B] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S)),R2)
     => ( ( S6 = S )
       => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S6)),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_5215_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),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))),R2),S2)) ) ).

% inf_Int_eq2
tff(fact_5216_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),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))),R2),S2)) ) ).

% sup_Un_eq2
tff(fact_5217_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_5218_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less(nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_5219_sorted__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,Na)) ).

% sorted_upt
tff(fact_5220_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xsa))
        <=> sorted_wrt(B,aTP_Lamp_oc(fun(B,A),fun(B,fun(B,$o)),F2),Xsa) ) ) ).

% sorted_map
tff(fact_5221_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_5222_sorted__wrt01,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
     => sorted_wrt(A,P,Xsa) ) ).

% sorted_wrt01
tff(fact_5223_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xsa: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xsa),I)),aa(nat,A,nth(A,Xsa),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5224_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( sorted_wrt(A,P,Xsa)
    <=> ! [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)),Xsa))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xsa),I4)),aa(nat,A,nth(A,Xsa),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5225_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ysa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( distinct(A,Xsa)
           => ( sorted_wrt(A,ord_less_eq(A),Ysa)
             => ( distinct(A,Ysa)
               => ( ( aa(list(A),set(A),set2(A),Xsa) = aa(list(A),set(A),set2(A),Ysa) )
                 => ( Xsa = Ysa ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5226_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xsa) ) ) ).

% sorted01
tff(fact_5227_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I4)),aa(nat,A,nth(A,Xsa),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5228_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ? [X4: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X4) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X4)
              & distinct(A,X4)
              & ! [Y2: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y2) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y2)
                    & distinct(A,Y2) )
                 => ( Y2 = X4 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5229_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_5230_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I4)),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5231_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I)),aa(nat,A,nth(A,Xsa),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_5232_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I4)),aa(nat,A,nth(A,Xsa),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5233_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5234_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A12))
     => ( ! [K2: int,L2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( aa(set(int),$o,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & aa(set(int),$o,member(int,L2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A12) ) ) ).

% and_int.pinduct
tff(fact_5235_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R3: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),one_one(A))),aa(A,A,minus_minus(A,R3),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),R3)) ) ).

% divmod_step_eq
tff(fact_5236_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa($o,int,zero_neq_one_of_bool(int),R3 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_5237_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5238_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Y) ).

% bot2E
tff(fact_5239_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_5240_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N))
       => ( ! [N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))
         => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),one2)
           => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),bit0(N))
             => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),aa(num,num,bit1,N))
               => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2)
                 => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),bit0(N))
                   => ~ ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5241_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_5242_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5243_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R3 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
              $ite(R3 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),R3)))) ) ) ).

% zminus1_lemma
tff(fact_5244_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R3) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5245_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),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),Q3),L)),R3) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5246_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
     => ( ( ( A23 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A12) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
             => ( ( A23 != zero_zero(int) )
               => ( A12 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A23) ) ) )
         => ~ ! [R: int,Q5: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R) )
               => ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),A23) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),A23))
                   => ( A12 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A23)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5247_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5248_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5249_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A12))
     => ( ! [I2: int,J2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A12) ) ) ).

% upto.pinduct
tff(fact_5250_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,Na),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,Na))))) ) ).

% divmod_algorithm_code(8)
tff(fact_5251_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,aa(num,num,bit1,Na),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,Na))))) ) ).

% divmod_algorithm_code(7)
tff(fact_5252_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Na))) ).

% minus_numeral_div_numeral
tff(fact_5253_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Na))) ).

% numeral_div_minus_numeral
tff(fact_5254_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_5255_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,bit0(Na)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_5256_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,Na)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_5257_minus__one__div__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Na))) ).

% minus_one_div_numeral
tff(fact_5258_one__div__minus__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Na))) ).

% one_div_minus_numeral
tff(fact_5259_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(nat,M,Na) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Na))) ).

% divmod'_nat_def
tff(fact_5260_divmod__int__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(int,M,Na) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na))) ).

% divmod_int_def
tff(fact_5261_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,M,Na) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Na))) ) ).

% divmod_def
tff(fact_5262_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,M,Na) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,Na,unique8689654367752047608divmod(A,M,bit0(Na)))) ) ).

% divmod_divmod_step
tff(fact_5263_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Na))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,Na,M)) ) ) ).

% dvd_numeral_simp
tff(fact_5264_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_5265_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xsa: list(A),Ysa: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ysa)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xsa,Ysa)),Na) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(list(B),nat,size_size(list(B)),Ysa)))),aa(nat,B,nth(B,Ysa),modulo_modulo(nat,Na,aa(list(B),nat,size_size(list(B)),Ysa)))) ) ) ).

% product_nth
tff(fact_5266_length__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ysa: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xsa,Ysa)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ysa)) ).

% length_product
tff(fact_5267_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ysa: list(B)] :
      ( distinct(A,Xsa)
     => ( distinct(B,Ysa)
       => distinct(product_prod(A,B),product(A,B,Xsa,Ysa)) ) ) ).

% distinct_product
tff(fact_5268_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F2: fun(A,nat)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_5269_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc))) )
           => ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_5270_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc))))
     => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc,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),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_5271_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A12: nat,A23: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A12),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A23),A32))))
     => ( ! [F4: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc2: A] :
            ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F4),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc2))))
           => ( ( ~ 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,F4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F4,A4),Acc2)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F4),A4),B4),Acc2) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A12),A23),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_5272_in__finite__psubset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A3),B3)),finite_psubset(A))
    <=> ( 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_5273_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_5274_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(Na)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_od(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(6)
tff(fact_5275_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_5276_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_5277_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,bit0(M),bit0(Na)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_oe(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(5)
tff(fact_5278_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X4: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) = Q3 )
         => ( aa(B,C,aa(A,fun(B,C),F2,X4),Y3) = aa(B,C,aa(A,fun(B,C),G,X4),Y3) ) )
     => ( ( P3 = Q3 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q3) ) ) ) ).

% split_cong
tff(fact_5279_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5280_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(bit0(Na)) = aa(num,num,bit1,bitM(Na)) ).

% semiring_norm(27)
tff(fact_5281_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(aa(num,num,bit1,Na)) = aa(num,num,bit1,bit0(Na)) ).

% semiring_norm(28)
tff(fact_5282_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_5283_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = aa(num,num,bit1,Na) ).

% BitM_inc_eq
tff(fact_5284_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),bit0(Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Na))) ).

% eval_nat_numeral(2)
tff(fact_5285_one__plus__BitM,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Na)) = bit0(Na) ).

% one_plus_BitM
tff(fact_5286_BitM__plus__one,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Na)),one2) = bit0(Na) ).

% BitM_plus_one
tff(fact_5287_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bitM(Na)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(Na))),one_one(A)) ) ).

% numeral_BitM
tff(fact_5288_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W2: num] : ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(num,A,numeral_numeral(A),bitM(W2))) ) ).

% odd_numeral_BitM
tff(fact_5289_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(Na))) ) ).

% not_numeral_BitM_eq
tff(fact_5290_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_of(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5291_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_og(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5292_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_oh(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5293_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F2,A2),B2) ).

% case_prod_conv
tff(fact_5294_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_oi(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_5295_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)
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_5296_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(set(C),$o,member(C,Z2),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B4)) )
     => aa(set(C),$o,member(C,Z2),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P3)) ) ).

% mem_case_prodI2
tff(fact_5297_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( aa(set(A),$o,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2))
     => aa(set(A),$o,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2))) ) ).

% mem_case_prodI
tff(fact_5298_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),X: C] :
      ( ! [A4: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = P3 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B4),X) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),X) ) ).

% case_prodI2'
tff(fact_5299_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A4: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B4) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3) ) ).

% case_prodI2
tff(fact_5300_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A2),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_5301_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_oj(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_5302_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_oj(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_5303_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P3: product_prod(A,B),Z2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),Z2)
     => ~ ! [X4: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X4),Y3),Z2) ) ) ).

% case_prodE'
tff(fact_5304_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R2,A2),B2),C2) ) ).

% case_prodD'
tff(fact_5305_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3)
     => ~ ! [X4: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X4),Y3) ) ) ).

% case_prodE
tff(fact_5306_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F2,A2),B2) ) ).

% case_prodD
tff(fact_5307_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ok(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_5308_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A3),B3)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A3))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B3))) ) ).

% Collect_case_prod_mono
tff(fact_5309_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ol(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_on(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_5310_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ol(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_op(fun(nat,fun(nat,A)),fun(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_5311_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_oq(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_on(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sum.triangle_reindex
tff(fact_5312_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_oq(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_op(fun(nat,fun(nat,A)),fun(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.triangle_reindex
tff(fact_5313_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),Xsa: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xsa))
       => distinct(B,Xsa) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_5314_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_or(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_5315_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D6: $tType,C: $tType,H: fun(B,A),F2: fun(C,fun(D6,B)),Prod: product_prod(C,D6)] : aa(B,A,H,aa(product_prod(C,D6),B,aa(fun(C,fun(D6,B)),fun(product_prod(C,D6),B),product_case_prod(C,D6,B),F2),Prod)) = aa(product_prod(C,D6),A,aa(fun(C,fun(D6,A)),fun(product_prod(C,D6),A),product_case_prod(C,D6,A),aa(fun(C,fun(D6,B)),fun(C,fun(D6,A)),aTP_Lamp_os(fun(B,A),fun(fun(C,fun(D6,B)),fun(C,fun(D6,A))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_5316_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_ot(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5317_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2))
     => ~ ! [X4: B,Y3: C] :
            ( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y3) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5318_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ou(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_5319_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F2,X4),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_5320_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,semiring_1_of_nat(A),Na) = $ite(Na = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_ov(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5321_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_5322_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_ow(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than2_def
tff(fact_5323_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ox($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X3)
    <=> ( (P)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X3) ) ) ).

% split_part
tff(fact_5324_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_oy(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_5325_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_5326_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_5327_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_5328_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_5329_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_5330_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_5331_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_5332_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F2: fun(B,A),Xsa: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xsa)),S2)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),Xsa))
         => ( distinct(B,Xsa)
           => ( 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),Xsa)) = Xsa ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_5333_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_oz(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than_def
tff(fact_5334_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,$o)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2)
       => ( aa(set(B),$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)),insert(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_5335_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,$o)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( ~ aa(set(B),$o,member(B,X),A3)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_5336_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_5337_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pa(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_5338_case__prod__app,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D6,A))),X: product_prod(B,C),Y: D6] : aa(D6,A,aa(product_prod(B,C),fun(D6,A),aa(fun(B,fun(C,fun(D6,A))),fun(product_prod(B,C),fun(D6,A)),product_case_prod(B,C,fun(D6,A)),F2),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_pb(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),F2),Y)),X) ).

% case_prod_app
tff(fact_5339_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,$o)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2)
       => ( aa(set(B),$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),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B))))) = remove1(B,X,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_5340_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xsa: list(A)] :
      ( ( A2 != B2 )
     => ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),remove1(A,B2,Xsa)))
      <=> aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% in_set_remove1
tff(fact_5341_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_5342_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_5343_set__remove1__eq,axiom,
    ! [A: $tType,Xsa: list(A),X: A] :
      ( distinct(A,Xsa)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xsa)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_5344_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_5345_distinct__remove1,axiom,
    ! [A: $tType,Xsa: list(A),X: A] :
      ( distinct(A,Xsa)
     => distinct(A,remove1(A,X,Xsa)) ) ).

% distinct_remove1
tff(fact_5346_remove1__commute,axiom,
    ! [A: $tType,X: A,Y: A,Zs: list(A)] : remove1(A,X,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,X,Zs)) ).

% remove1_commute
tff(fact_5347_remove1__idem,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( remove1(A,X,Xsa) = Xsa ) ) ).

% remove1_idem
tff(fact_5348_notin__set__remove1,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),remove1(A,Y,Xsa))) ) ).

% notin_set_remove1
tff(fact_5349_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% set_remove1_subset
tff(fact_5350_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xsa)) ) ) ).

% sorted_remove1
tff(fact_5351_quotient__of__denom__pos,axiom,
    ! [R3: rat,P3: int,Q3: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).

% quotient_of_denom_pos
tff(fact_5352_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xsa))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),remove1(B,X,Xsa))) ) ) ).

% sorted_map_remove1
tff(fact_5353_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),R2: $o,X: A,Y: B] :
      ( ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
        & (R2) )
     => ( (R2)
        & ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
         => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ) ).

% predicate2D_conj
tff(fact_5354_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X4: fun(A,B)] : aa(fun(A,B),C,F2,X4) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_pc(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_5355_length__remove1,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,X,Xsa)) = $ite(aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa)),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_remove1
tff(fact_5356_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [X: A,Xsa: list(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xsa)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),remove1(A,X,Xsa)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_5357_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_pd(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_5358_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pe(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_uminus_code
tff(fact_5359_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q3)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_pg(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_5360_rat__floor__code,axiom,
    ! [P3: rat] : archim6421214686448440834_floor(rat,P3) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P3)) ).

% rat_floor_code
tff(fact_5361_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ph(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_abs_code
tff(fact_5362_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P3),Q3)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_pj(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_5363_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).

% case_prod_Pair_iden
tff(fact_5364_rat__plus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pl(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_5365_rat__minus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,minus_minus(rat,P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pn(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_5366_VEBT__internal_Ofoldr__zero,axiom,
    ! [Xsa: list(nat),D3: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xsa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xsa),I2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xsa)),aa(nat,nat,minus_minus(nat,aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xsa),D3)),D3)) ) ).

% VEBT_internal.foldr_zero
tff(fact_5367_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_5368_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q3),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_5369_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X4: B] :
              ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X4),A4) = aa(A,A,aa(B,fun(A,A),G,X4),A4) ) )
         => ( aa(A,A,foldr(B,A,F2,L),A2) = aa(A,A,foldr(B,A,G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5370_VEBT__internal_Ofoldr__one,axiom,
    ! [D3: nat,Ysa: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D3),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ysa),D3)) ).

% VEBT_internal.foldr_one
tff(fact_5371_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xsa: list(A)] : groups8242544230860333062m_list(A,Xsa) = aa(A,A,foldr(A,A,plus_plus(A),Xsa),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_5372_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).

% normalize_denom_pos
tff(fact_5373_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P3: int,R3: int] :
      ( ( Q3 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R3),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R3),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_5374_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xsa: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xsa) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_po(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xsa),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_5375_VEBT__internal_Ofoldr__same__int,axiom,
    ! [Xsa: list(nat),Y: nat] :
      ( ! [X4: nat,Y3: nat] :
          ( aa(set(nat),$o,member(nat,X4),aa(list(nat),set(nat),set2(nat),Xsa))
         => ( aa(set(nat),$o,member(nat,Y3),aa(list(nat),set(nat),set2(nat),Xsa))
           => ( X4 = Y3 ) ) )
     => ( ! [X4: nat] :
            ( aa(set(nat),$o,member(nat,X4),aa(list(nat),set(nat),set2(nat),Xsa))
           => ( X4 = Y ) )
       => ( aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xsa),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xsa)),Y) ) ) ) ).

% VEBT_internal.foldr_same_int
tff(fact_5376_VEBT__internal_Ofoldr__mono,axiom,
    ! [Xsa: list(nat),Ysa: list(nat),C2: nat,D3: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xsa) = aa(list(nat),nat,size_size(list(nat)),Ysa) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xsa))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,nth(nat,Xsa),I2)),aa(nat,nat,nth(nat,Ysa),I2)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),D3)
         => 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,foldr(nat,nat,plus_plus(nat),Xsa),C2)),aa(list(nat),nat,size_size(list(nat)),Ysa))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ysa),D3)) ) ) ) ).

% VEBT_internal.foldr_mono
tff(fact_5377_VEBT__internal_Olist__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X4)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),Xsa)),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Bound)),I)) ) ).

% VEBT_internal.list_every_elemnt_bound_sum_bound
tff(fact_5378_rat__times__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pq(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_times_code
tff(fact_5379_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ps(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_5380_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList)),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_5381_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_5382_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList)),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_5383_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary))),aa(real,real,foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList)),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5384_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_5385_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_5386_VEBT__internal_Oheight__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ).

% VEBT_internal.height_compose_summary
tff(fact_5387_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_5388_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_5389_VEBT__internal_Olistsum__bound,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,real),Y: real] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X4)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xsa)),Y)) ) ).

% VEBT_internal.listsum_bound
tff(fact_5390_VEBT__internal_Ofoldr__same,axiom,
    ! [Xsa: list(real),Y: real] :
      ( ! [X4: real,Y3: real] :
          ( aa(set(real),$o,member(real,X4),aa(list(real),set(real),set2(real),Xsa))
         => ( aa(set(real),$o,member(real,Y3),aa(list(real),set(real),set2(real),Xsa))
           => ( X4 = Y3 ) ) )
     => ( ! [X4: real] :
            ( aa(set(real),$o,member(real,X4),aa(list(real),set(real),set2(real),Xsa))
           => ( X4 = Y ) )
       => ( aa(real,real,foldr(real,real,plus_plus(real),Xsa),zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xsa))),Y) ) ) ) ).

% VEBT_internal.foldr_same
tff(fact_5391_VEBT__internal_Oheight__compose__child,axiom,
    ! [T2: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summary: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,T2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ) ).

% VEBT_internal.height_compose_child
tff(fact_5392_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_5393_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_5394_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_5395_VEBT__internal_Of__g__map__foldr__bound,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D3: real] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X4)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X4))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xsa)),D3)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,G),Xsa)),zero_zero(real)))),D3)) ) ).

% VEBT_internal.f_g_map_foldr_bound
tff(fact_5396_VEBT__internal_Oreal__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),Xsa)),C2)) = aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,aTP_Lamp_pt(fun(A,nat),fun(A,real),F2)),Xsa)),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% VEBT_internal.real_nat_list
tff(fact_5397_VEBT__internal_Olist__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X4)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xsa)),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xsa))),Bound)),I)) ) ).

% VEBT_internal.list_every_elemnt_bound_sum_bound_real
tff(fact_5398_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_5399_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_VEBT_insert(vEBT_Node(Info,Deg,TreeList,Summary),X) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)),X),vEBT_Node(Info,Deg,TreeList,Summary),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_5400_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),X) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_5401_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5402_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList)),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5403_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_5404_VEBT__internal_OLeaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% VEBT_internal.Leaf_0_not
tff(fact_5405_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A2),(B2))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_5406_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_5407_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X4)
     => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Ux)
       => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_5408_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_5409_vebt__delete__code_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),X) = $ite(
        X = zero_zero(nat),
        vEBT_Leaf($false,(B2)),
        $ite(X = one_one(nat),vEBT_Leaf((A2),$false),vEBT_Leaf((A2),(B2))) ) ).

% vebt_delete_code(1)
tff(fact_5410_vebt__insert__code_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Leaf((A2),(B2))),X) = $ite(
        X = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(X = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert_code(1)
tff(fact_5411_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_5412_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_5413_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_5414_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_5415_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_5416_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_5417_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,X) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5418_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_5419_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_5420_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_5421_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_5422_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_5423_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(X = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_5424_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_5425_VEBT__internal_Ocnt__cnt__eq,axiom,
    ! [T2: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,T2)) ).

% VEBT_internal.cnt_cnt_eq
tff(fact_5426_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_5427_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),aa(real,real,foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2)),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5428_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(X = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_5429_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( Y != aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Leaf((A4),(B4))),Xa) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa),vEBT_Node(Info2,Deg2,TreeList2,Summary2),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),Xa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_5430_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5431_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info,zero_zero(nat),Ts,S)),X) = vEBT_Node(Info,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_5432_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2)),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_5433_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2)),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_5434_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Leaf((A4),(B4))),Xa) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa),vEBT_Node(Info2,Deg2,TreeList2,Summary2),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info2,Deg2,TreeList2,Summary2)),Xa)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,Deg2,TreeList2,Summary2)),Xa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_5435_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(real) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),aa(real,real,foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2)),zero_zero(real))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_5436_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2)),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_5437_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf((A2),$false) ).

% vebt_delete.simps(2)
tff(fact_5438_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = vEBT_Leaf($false,(B2)) ).

% vebt_delete.simps(1)
tff(fact_5439_VEBT__internal_Omi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% VEBT_internal.mi_ma_2_deg
tff(fact_5440_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_5441_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_5442_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: nat,F2: fun(A,nat)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,X))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(A),nat,size_list(A,F2),Xsa)) ) ) ).

% size_list_estimation
tff(fact_5443_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: nat,F2: fun(A,nat)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,X))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F2),Xsa)) ) ) ).

% size_list_estimation'
tff(fact_5444_size__list__pointwise,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X4)),aa(A,nat,G,X4)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_list(A,F2),Xsa)),aa(list(A),nat,size_list(A,G),Xsa)) ) ).

% size_list_pointwise
tff(fact_5445_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_5446_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_T_p_r_e_d(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_5447_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_5448_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_T_p_r_e_d2(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_5449_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c2(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_5450_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Uy2,Uz),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_5451_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_5452_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_T_p_r_e_d(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_5453_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_T_s_u_c_c(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_5454_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_T_p_r_e_d2(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_5455_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_T_s_u_c_c2(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_5456_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),X) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_5457_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),X) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_5458_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_5459_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Uy2,Uz),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_5460_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),X) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_5461_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),X) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_5462_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary))) ) ).

% height_node
tff(fact_5463_size__list__conv__sum__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : aa(list(A),nat,size_list(A,F2),Xsa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% size_list_conv_sum_list
tff(fact_5464_VEBT__internal_Oinsert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% VEBT_internal.insert_simp_mima
tff(fact_5465_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_5466_VEBT__internal_Odelt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% VEBT_internal.delt_out_of_range
tff(fact_5467_VEBT__internal_Otdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% VEBT_internal.tdeletemimi
tff(fact_5468_VEBT__internal_Otdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),X)),one_one(nat)) ) ).

% VEBT_internal.tdeletemimi'
tff(fact_5469_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_5470_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),some(A,X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_5471_VEBT__internal_Odel__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% VEBT_internal.del_single_cont
tff(fact_5472_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_5473_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_5474_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_5475_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_5476_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_5477_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Uw)
     => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X4)
         => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),X4)
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_5478_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))
     => ( ! [A4: $o,B4: $o] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B4: $o,N: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Uu)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),X4)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),X4)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X4) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_5479_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X4)
     => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),X4)
       => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)),X4)
         => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X4) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_5480_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X4)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S3)),X4)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)),X4)
         => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X4) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_5481_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,B4: $o] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))
     => ( ! [Uv: $o,Uw: $o,N: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))
       => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Va3)
         => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Ve2)
           => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X4) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_5482_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))
     => ( ! [A4: $o,Uw: $o] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B4: $o,Va: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))
         => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Vb2)
           => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)),Vf2)
             => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X4) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_5483_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5484_set__vebt__maxt_H,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_maxt(T2) = some(nat,X) )
      <=> ( aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2))
          & ! [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),X) ) ) ) ) ).

% set_vebt_maxt'
tff(fact_5485_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(X) = Y )
     => ( ! [A4: $o] :
            ( ? [B4: $o] : X = vEBT_Leaf((A4),(B4))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_5486_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,Na: nat,Y: A] :
      ( ( replicate(A,M,X) = replicate(A,Na,Y) )
    <=> ( ( M = Na )
        & ( ( M != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_5487_length__replicate,axiom,
    ! [A: $tType,Na: nat,X: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Na,X)) = Na ).

% length_replicate
tff(fact_5488_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Na: nat,X: B] : aa(list(B),list(A),map(B,A,F2),replicate(B,Na,X)) = replicate(A,Na,aa(B,A,F2,X)) ).

% map_replicate
tff(fact_5489_Ball__set__replicate,axiom,
    ! [A: $tType,Na: nat,A2: A,P: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),replicate(A,Na,A2)))
         => aa(A,$o,P,X2) )
    <=> ( aa(A,$o,P,A2)
        | ( Na = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_5490_Bex__set__replicate,axiom,
    ! [A: $tType,Na: nat,A2: A,P: fun(A,$o)] :
      ( ? [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),replicate(A,Na,A2)))
          & aa(A,$o,P,X2) )
    <=> ( aa(A,$o,P,A2)
        & ( Na != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_5491_in__set__replicate,axiom,
    ! [A: $tType,X: A,Na: nat,Y: A] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),replicate(A,Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_5492_nth__replicate,axiom,
    ! [A: $tType,I: nat,Na: nat,X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,replicate(A,Na,X)),I) = X ) ) ).

% nth_replicate
tff(fact_5493_set__replicate,axiom,
    ! [A: $tType,Na: nat,X: A] :
      ( ( Na != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Na,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_5494_set__vebt__maxt,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_maxt(T2) = some(nat,X) )
      <=> vEBT_VEBT_max_in_set(vEBT_set_vebt(T2),X) ) ) ).

% set_vebt_maxt
tff(fact_5495_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A2),(B2))) = $ite(
        (B2),
        some(nat,one_one(nat)),
        $ite((A2),some(nat,zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_5496_VEBT__internal_Ointind,axiom,
    ! [A: $tType,I: nat,Na: nat,P: fun(A,$o),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(A,$o,P,X)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Na,X)),I)) ) ) ).

% VEBT_internal.intind
tff(fact_5497_replicate__length__same,axiom,
    ! [A: $tType,Xsa: list(A),X: A] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => ( X4 = X ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xsa),X) = Xsa ) ) ).

% replicate_length_same
tff(fact_5498_replicate__eqI,axiom,
    ! [A: $tType,Xsa: list(A),Na: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = Na )
     => ( ! [Y3: A] :
            ( aa(set(A),$o,member(A,Y3),aa(list(A),set(A),set2(A),Xsa))
           => ( Y3 = X ) )
       => ( Xsa = replicate(A,Na,X) ) ) ) ).

% replicate_eqI
tff(fact_5499_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Na: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Na,X)) ) ).

% sorted_replicate
tff(fact_5500_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_pu(A,fun(B,A)),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_5501_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat,C2: A] : groups8242544230860333062m_list(A,replicate(A,Na,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),C2) ) ).

% sum_list_replicate
tff(fact_5502_vebt__inst_Oset__vebt__maxt_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = some(nat,X) )
      <=> ( aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2))
          & ! [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),X) ) ) ) ) ).

% vebt_inst.set_vebt_maxt'
tff(fact_5503_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  some(nat,one_one(nat)),
                  $ite((A4),some(nat,zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != some(nat,Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_5504_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_pv(A,fun(nat,A),X)),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_5505_set__replicate__Suc,axiom,
    ! [A: $tType,Na: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Na),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_5506_set__replicate__conv__if,axiom,
    ! [A: $tType,Na: nat,X: A] :
      aa(list(A),set(A),set2(A),replicate(A,Na,X)) = $ite(Na = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_5507_mint__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(T2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_5508_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_5509_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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.simps(3)
tff(fact_5510_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),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)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5511_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_5512_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (B4),
                      some(nat,one_one(nat)),
                      $ite((A4),some(nat,zero_zero(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = some(nat,Ma2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_5513_VEBT__internal_Omax__in__set__def,axiom,
    ! [Xsa: set(nat),X: nat] :
      ( vEBT_VEBT_max_in_set(Xsa,X)
    <=> ( aa(set(nat),$o,member(nat,X),Xsa)
        & ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),Xsa)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),X) ) ) ) ).

% VEBT_internal.max_in_set_def
tff(fact_5514_vebt__inst_Oset__vebt__maxt,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = some(nat,X) )
      <=> vEBT_VEBT_max_in_set(vEBT_set_vebt(T2),X) ) ) ).

% vebt_inst.set_vebt_maxt
tff(fact_5515_set__vebt__mint_H,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_mint(T2) = some(nat,X) )
      <=> ( aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2))
          & ! [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),X2) ) ) ) ) ).

% set_vebt_mint'
tff(fact_5516_VEBT__internal_Omintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Ma)
          & ? [M4: nat] :
              ( ( some(nat,M4) = vEBT_vebt_mint(Summary) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% VEBT_internal.mintlistlength
tff(fact_5517_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X22: A] : size_option(A,X,some(A,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_5518_vebt__minNull__mint,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T2)
    <=> ( vEBT_vebt_mint(T2) = none(nat) ) ) ).

% vebt_minNull_mint
tff(fact_5519_vebt__inst_Oset__vebt__mint_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = some(nat,X) )
      <=> ( aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2))
          & ! [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),X2) ) ) ) ) ).

% vebt_inst.set_vebt_mint'
tff(fact_5520_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A2),(B2))) = $ite(
        (A2),
        some(nat,zero_zero(nat)),
        $ite((B2),some(nat,one_one(nat)),none(nat)) ) ).

% vebt_mint.simps(1)
tff(fact_5521_VEBT__internal_Omisiz,axiom,
    ! [T2: vEBT_VEBT,Na: nat,M: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( some(nat,M) = vEBT_vebt_mint(T2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% VEBT_internal.misiz
tff(fact_5522_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_option(A,X,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_5523_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  some(nat,zero_zero(nat)),
                  $ite((B4),some(nat,one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != some(nat,Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_5524_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (A4),
                      some(nat,zero_zero(nat)),
                      $ite((B4),some(nat,one_one(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = some(nat,Mi2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_5525_set__vebt__mint,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_mint(T2) = some(nat,X) )
      <=> vEBT_VEBT_min_in_set(vEBT_set_vebt(T2),X) ) ) ).

% set_vebt_mint
tff(fact_5526_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X4,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_5527_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(T2)) ).

% set_vebt_def
tff(fact_5528_VEBT__internal_Omin__in__set__def,axiom,
    ! [Xsa: set(nat),X: nat] :
      ( vEBT_VEBT_min_in_set(Xsa,X)
    <=> ( aa(set(nat),$o,member(nat,X),Xsa)
        & ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),Xsa)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),X2) ) ) ) ).

% VEBT_internal.min_in_set_def
tff(fact_5529_insersimp_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_5530_insersimp,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_5531_VEBT__internal_Ovalid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)),X) ) ) ).

% VEBT_internal.valid_insert_both_member_options_add
tff(fact_5532_VEBT__internal_Ovalid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( 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))),Na))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),Y)),X) ) ) ) ) ).

% VEBT_internal.valid_insert_both_member_options_pres
tff(fact_5533_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X4,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_5534_vebt__inst_Oset__vebt__mint,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = some(nat,X) )
      <=> vEBT_VEBT_min_in_set(vEBT_set_vebt(T2),X) ) ) ).

% vebt_inst.set_vebt_mint
tff(fact_5535_set__vebt__mint_H_H,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_vebt_mint(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),some(nat,aa(set(nat),nat,lattic643756798350308766er_Min(nat),vEBT_set_vebt(T2)))) ) ) ).

% set_vebt_mint''
tff(fact_5536_set__vebt__maxt_H_H,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_vebt_maxt(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),some(nat,aa(set(nat),nat,lattic643756798349783984er_Max(nat),vEBT_set_vebt(T2)))) ) ) ).

% set_vebt_maxt''
tff(fact_5537_VEBT__internal_Opower__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa(nat,nat,power_power(nat,X),Y) = Z2 )
    <=> ( vEBT_VEBT_power(some(nat,X),some(nat,Y)) = some(nat,Z2) ) ) ).

% VEBT_internal.power_shift
tff(fact_5538_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_5539_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_5540_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ld(nat,fun(nat,$o),Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_5541_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5542_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),X) ) ) ) ) ) ).

% Max_less_iff
tff(fact_5543_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_5544_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X2) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_5545_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X) ) ) ) ).

% Min_le
tff(fact_5546_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [Y3: A] :
                ( aa(set(A),$o,member(A,Y3),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
           => ( aa(set(A),$o,member(A,X),A3)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_5547_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_5548_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A3) ) ) ) ).

% Min_in
tff(fact_5549_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max_ge
tff(fact_5550_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [Y3: A] :
                ( aa(set(A),$o,member(A,Y3),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
           => ( aa(set(A),$o,member(A,X),A3)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_5551_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),$o,finite_finite2(A),B3)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                 => ? [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),B3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa3) ) )
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),B3)
                   => ? [Xa3: A] :
                        ( aa(set(A),$o,member(A,Xa3),A3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa3) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B3) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5552_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max.coboundedI
tff(fact_5553_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3) ) ) ) ).

% Max_in
tff(fact_5554_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ).

% Min.boundedI
tff(fact_5555_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A11) ) ) ) ) ) ).

% Min.boundedE
tff(fact_5556_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A3) )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X2) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_5557_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
            <=> ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).

% Min_le_iff
tff(fact_5558_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = M )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X2) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_5559_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
            <=> ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),X) ) ) ) ) ) ).

% Min_less_iff
tff(fact_5560_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B4: A] :
                ( aa(set(A),$o,member(A,B4),A3)
               => aa(A,$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)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_5561_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$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,lattic643756798349783984er_Max(A),A3)),X) ) ) ) ) ).

% Max.boundedI
tff(fact_5562_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A11),X) ) ) ) ) ) ).

% Max.boundedE
tff(fact_5563_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_5564_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_5565_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = M )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_5566_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X2) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_5567_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B4: A] :
                ( aa(set(A),$o,member(A,B4),A3)
               => aa(A,$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)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_5568_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_5569_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N3)
         => ( ( M7 != bot_bot(set(A)) )
           => ( aa(set(A),$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),M7)) ) ) ) ) ).

% Min_antimono
tff(fact_5570_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_5571_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N3)
         => ( ( M7 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M7)),aa(set(A),A,lattic643756798349783984er_Max(A),N3)) ) ) ) ) ).

% Max_mono
tff(fact_5572_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_5573_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(Na = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pw(nat,fun(nat,fun(nat,$o)),M),Na)))) ).

% divide_nat_def
tff(fact_5574_vebt__inst_Oset__vebt__maxt_H_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_maxt(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),some(nat,aa(set(nat),nat,lattic643756798349783984er_Max(nat),vEBT_set_vebt(T2)))) ) ) ).

% vebt_inst.set_vebt_maxt''
tff(fact_5575_vebt__inst_Oset__vebt__mint_H_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_mint(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),some(nat,aa(set(nat),nat,lattic643756798350308766er_Min(nat),vEBT_set_vebt(T2)))) ) ) ).

% vebt_inst.set_vebt_mint''
tff(fact_5576_VEBT__internal_Osetceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Na: nat] :
      ( vEBT_invar_vebt(S,M)
     => ( ! [X4: vEBT_VEBT] :
            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X4,Na) )
       => ( ( M = aa(nat,nat,suc,Na) )
         => ( ! [X4: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X4)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,S)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),S),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% VEBT_internal.setceilmax
tff(fact_5577_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xsa))
                & aa(A,$o,P,X3) )
           => ( find(A,P,Xsa) = some(A,aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_px(list(A),fun(fun(A,$o),fun(A,$o)),Xsa),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_5578_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
     => ( ( ? [A4: $o,B4: $o] : A12 = vEBT_Leaf((A4),(B4))
         => ( A23 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                     => vEBT_invar_vebt(X3,N) )
                 => ( vEBT_invar_vebt(Summary2,M4)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                     => ( ( M4 = N )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                         => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                       => vEBT_invar_vebt(X3,N) )
                   => ( vEBT_invar_vebt(Summary2,M4)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                       => ( ( M4 = aa(nat,nat,suc,N) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                           => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_invar_vebt(X3,N) )
                     => ( vEBT_invar_vebt(Summary2,M4)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                         => ( ( M4 = N )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                             => ( ! [I3: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                 => ( 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))),Deg2))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N) = I3 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N) = I3 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_invar_vebt(X3,N) )
                       => ( vEBT_invar_vebt(Summary2,M4)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                           => ( ( M4 = aa(nat,nat,suc,N) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                               => ( ! [I3: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                         => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N) = I3 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N) = I3 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_5579_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B,A3: set(B)] :
      ( ( B2 = aa(B,A,F2,X) )
     => ( aa(set(B),$o,member(B,X),A3)
       => aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),A3)) ) ) ).

% image_eqI
tff(fact_5580_image__ident,axiom,
    ! [A: $tType,Y4: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ab(A,A)),Y4) = Y4 ).

% image_ident
tff(fact_5581_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_5582_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F2),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_5583_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_5584_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B3: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F2,A2)),aa(set(B),set(A),image(B,A,F2),B3)) ).

% image_insert
tff(fact_5585_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,member(A,X),A3)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,X)),aa(set(A),set(B),image(A,B,F2),A3)) = aa(set(A),set(B),image(A,B,F2),A3) ) ) ).

% insert_image
tff(fact_5586_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_5587_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),image(B,A,F2),aa(list(B),set(B),set2(B),V2)) ).

% list.set_map
tff(fact_5588_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_py(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_5589_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_pz(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A2),D3),aa(A,A,minus_minus(A,B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5590_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_py(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_5591_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),image(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qa(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_qb(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_5592_Max__divisors__self__int,axiom,
    ! [Na: int] :
      ( ( Na != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_mm(int,fun(int,$o),Na))) = aa(int,int,abs_abs(int),Na) ) ) ).

% Max_divisors_self_int
tff(fact_5593_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),image(A,B,aTP_Lamp_qc(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_5594_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),image(A,B,aTP_Lamp_qc(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Min_const
tff(fact_5595_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5596_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_qd(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5597_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),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_5598_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),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_5599_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : aa(set(B),set(A),image(B,A,F2),aa(list(B),set(B),set2(B),Xsa)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),Xsa)) ).

% image_set
tff(fact_5600_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),image(A,B,F2),A3))
       => aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% finite_surj
tff(fact_5601_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),image(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),image(B,A,F2),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_5602_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ? [B12: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B12)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B12),aa(set(B),set(A),image(B,A,F2),A3))
          & aa(set(A),$o,P,B12) )
    <=> ? [B12: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B12)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B12),A3)
          & aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F2),B12)) ) ) ).

% ex_finite_subset_image
tff(fact_5603_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B12: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B12)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B12),aa(set(B),set(A),image(B,A,F2),A3)) )
         => aa(set(A),$o,P,B12) )
    <=> ! [B12: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B12)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B12),A3) )
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F2),B12)) ) ) ).

% all_finite_subset_image
tff(fact_5604_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3)) ).

% image_Un
tff(fact_5605_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B12: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B12),aa(set(B),set(A),image(B,A,F2),A3))
         => aa(set(A),$o,P,B12) )
    <=> ! [B12: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B12),A3)
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F2),B12)) ) ) ).

% all_subset_image
tff(fact_5606_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B3: set(B)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(set(B),$o,member(B,aa(A,B,F2,X4)),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),aa(fun(A,$o),set(A),collect(A),P))),B3) ) ).

% image_Collect_subsetI
tff(fact_5607_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),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F2),A3))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
          & ( B3 = aa(set(B),set(A),image(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_5608_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),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B3)
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => aa(set(A),$o,member(A,aa(B,A,F2,X2)),B3) ) ) ).

% image_subset_iff
tff(fact_5609_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),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F2),A3))
     => ~ ! [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
           => ( B3 != aa(set(B),set(A),image(B,A,F2),C8) ) ) ) ).

% subset_imageE
tff(fact_5610_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(B),$o,member(B,aa(A,B,F2,X4)),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B3) ) ).

% image_subsetI
tff(fact_5611_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),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B3)) ) ).

% image_mono
tff(fact_5612_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),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),pow2(B,A3))),pow2(A,B3)) ) ).

% image_Pow_mono
tff(fact_5613_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),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,F2),A3))
       => ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A3)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_qe(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F2),X4))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5614_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),A3))
     => ~ ! [X4: B] :
            ( ( B2 = aa(B,A,F2,X4) )
           => ~ aa(set(B),$o,member(B,X4),A3) ) ) ).

% imageE
tff(fact_5615_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_qf(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_5616_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_qg(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A3),P)) = aa(set(B),set(A),image(B,A,F2),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_qh(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_5617_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,member(A,X),A3)
     => aa(set(B),$o,member(B,aa(A,B,F2,X)),aa(set(A),set(B),image(A,B,F2),A3)) ) ).

% imageI
tff(fact_5618_image__iff,axiom,
    ! [A: $tType,B: $tType,Z2: A,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,member(A,Z2),aa(set(B),set(A),image(B,A,F2),A3))
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
          & ( Z2 = aa(B,A,F2,X2) ) ) ) ).

% image_iff
tff(fact_5619_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(set(B),set(A),image(B,A,F2),A3))
          & aa(A,$o,P,X3) )
     => ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),A3)
          & aa(A,$o,P,aa(B,A,F2,X4)) ) ) ).

% bex_imageD
tff(fact_5620_image__cong,axiom,
    ! [B: $tType,A: $tType,M7: set(A),N3: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M7 = N3 )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),N3)
           => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
       => ( aa(set(A),set(B),image(A,B,F2),M7) = aa(set(A),set(B),image(A,B,G),N3) ) ) ) ).

% image_cong
tff(fact_5621_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(set(B),set(A),image(B,A,F2),A3))
         => aa(A,$o,P,X4) )
     => ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => aa(A,$o,P,aa(B,A,F2,X3)) ) ) ).

% ball_imageD
tff(fact_5622_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),B2: B,F2: fun(A,B)] :
      ( aa(set(A),$o,member(A,X),A3)
     => ( ( B2 = aa(A,B,F2,X) )
       => aa(set(B),$o,member(B,B2),aa(set(A),set(B),image(A,B,F2),A3)) ) ) ).

% rev_image_eqI
tff(fact_5623_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A3) = B3 )
     => ( aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),pow2(B,A3)) = pow2(A,B3) ) ) ).

% image_Pow_surj
tff(fact_5624_find__cong,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xsa = Ysa )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Ysa))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( find(A,P,Xsa) = find(A,Q,Ysa) ) ) ) ).

% find_cong
tff(fact_5625_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),minus_minus(set(A),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A3),B3))) ).

% image_diff_subset
tff(fact_5626_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),image(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),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3))) ).

% image_Int_subset
tff(fact_5627_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F6: fun(B,A),F2: fun(A,B),A7: set(B)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => ( aa(B,A,F6,aa(A,B,F2,X4)) = X4 ) )
     => ( ! [X4: B] :
            ( aa(set(B),$o,member(B,X4),A7)
           => ( aa(A,B,F2,aa(B,A,F6,X4)) = X4 ) )
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),A7)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F6),A7)),A3)
           => bij_betw(A,B,F2,A3,A7) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_5628_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A7: set(B),B3: set(A),B13: set(B)] :
      ( bij_betw(A,B,F2,A3,A7)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( ( aa(set(A),set(B),image(A,B,F2),B3) = B13 )
         => bij_betw(A,B,F2,B3,B13) ) ) ) ).

% bij_betw_subset
tff(fact_5629_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_pu(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)),insert(A),C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_5630_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),C2: B] :
      ( aa(set(A),$o,member(A,X),A3)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_qi(B,fun(A,B),C2)),A3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_5631_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_qk(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G)),aa(set(A),set(C),image(A,C,G),S2)) ) ) ) ).

% sum.image_gen
tff(fact_5632_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),S)),aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_Int
tff(fact_5633_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),aa(set(A),set(A),minus_minus(set(A),S),T2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),S)),aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_diff
tff(fact_5634_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)
         => ( groups7121269368397514597t_prod(A,B,H,S2) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_qm(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G),aa(set(A),set(C),image(A,C,G),S2)) ) ) ) ).

% prod.image_gen
tff(fact_5635_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_Compl
tff(fact_5636_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( aa(set(A),$o,member(A,Y3),A3)
           => ( aa(A,B,F2,Y3) = aa(A,B,F2,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F2),A3)) = aa(A,B,F2,X) ) ) ) ).

% the_elem_image_unique
tff(fact_5637_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),image(A,B,F2),A3))),aa(set(A),nat,finite_card(A),A3)) ) ).

% card_image_le
tff(fact_5638_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S2: set(A),T5: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),T5)
             => ( 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_qo(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H)),T5) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_5639_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( none(A) = find(A,P,Xsa) )
    <=> ~ ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
            & aa(A,$o,P,X2) ) ) ).

% find_None_iff2
tff(fact_5640_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( find(A,P,Xsa) = none(A) )
    <=> ~ ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
            & aa(A,$o,P,X2) ) ) ).

% find_None_iff
tff(fact_5641_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S2: set(A),T5: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),T5)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_qp(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H),T5) = groups7121269368397514597t_prod(A,C,H,S2) ) ) ) ) ) ).

% prod.group
tff(fact_5642_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),image(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_5643_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5644_VEBT__internal_Oboth__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),X) ) ) ) ).

% VEBT_internal.both_member_options_ding
tff(fact_5645_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),image(A,B,aa(B,fun(A,B),aTP_Lamp_qq(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),image(A,B,F2),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_5646_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),image(A,B,aa(B,fun(A,B),aTP_Lamp_qq(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),image(A,B,F2),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_5647_VEBT__internal_Oheight__compose__list,axiom,
    ! [T2: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,T2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ).

% VEBT_internal.height_compose_list
tff(fact_5648_VEBT__internal_Oboth__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% VEBT_internal.both_member_options_from_complete_tree_to_child
tff(fact_5649_VEBT__internal_Ohigh__def,axiom,
    ! [X: nat,Na: nat] : vEBT_VEBT_high(X,Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% VEBT_internal.high_def
tff(fact_5650_VEBT__internal_Olow__def,axiom,
    ! [X: nat,Na: nat] : vEBT_VEBT_low(X,Na) = modulo_modulo(nat,X,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% VEBT_internal.low_def
tff(fact_5651_VEBT__internal_Oboth__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X) ) ) ) ).

% VEBT_internal.both_member_options_from_chilf_to_complete_tree
tff(fact_5652_VEBT__internal_Ohigh__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Ma,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% VEBT_internal.high_bound_aux
tff(fact_5653_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,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),X),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),X),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5654_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_qr(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),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),X),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),X),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5655_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_qs(A,fun(A,fun(A,A)),M),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)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5656_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_qt(A,fun(A,fun(A,A)),M),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)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5657_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_qu(A,fun(A,fun(A,A)),M),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)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5658_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_qv(A,fun(A,fun(A,A)),M),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)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5659_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R2: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),R2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),R2)
             => ( 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_qw(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_qx(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G),F2)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5660_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),image(A,nat,F2),A3)))) ) ).

% sum_le_card_Max
tff(fact_5661_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),Xsa: list(A)] :
      ( ( some(A,X) = find(A,P,Xsa) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(A,$o,P,aa(nat,A,nth(A,Xsa),I4))
          & ( X = aa(nat,A,nth(A,Xsa),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xsa),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_5662_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A),X: A] :
      ( ( find(A,P,Xsa) = some(A,X) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(A,$o,P,aa(nat,A,nth(A,Xsa),I4))
          & ( X = aa(nat,A,nth(A,Xsa),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xsa),J3)) ) ) ) ).

% find_Some_iff
tff(fact_5663_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(X,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_5664_VEBT__internal_Olow__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),X),Na) = X ) ) ).

% VEBT_internal.low_inv
tff(fact_5665_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_5666_VEBT__internal_Ohigh__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),X),Na) = Y ) ) ).

% VEBT_internal.high_inv
tff(fact_5667_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X4,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X4,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X4)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_5668_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          X = Mi,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              X = Ma,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
                      one_one(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_5669_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          X = Mi,
          zero_zero(nat),
          $ite(
            X = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_5670_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),Mi,X),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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))
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_5671_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),image(A,nat,F2),A3)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ) ).

% card_Min_le_sum
tff(fact_5672_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X4,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X4,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X4)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_5673_VEBT__internal_Omax__ins__scaled,axiom,
    ! [Na: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))) ).

% VEBT_internal.max_ins_scaled
tff(fact_5674_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xa = Mi2,
                            zero_zero(nat),
                            $ite(
                              Xa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_5675_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $ite(
                            Xa = Mi2,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xa = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_5676_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero(nat),Ts2,S3)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_5677_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,X) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_5678_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
    <=> ( ( ? [A10: $o,B11: $o] : A12 = vEBT_Leaf((A10),(B11))
          & ( A23 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary3,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N2))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary3,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A23 = 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_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X2: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [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(Ma3,N2) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma3,N2)) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N2) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X2,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X2)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N2))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X2: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N2) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma3,N2)) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N2) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X2,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X2)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_5679_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S3)),Xa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)),Xa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xa = Mi2 )
                                      | ( Xa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_5680_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              $ite(
                                Xa = Mi2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xa = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_5681_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xa = Mi2,
                                zero_zero(nat),
                                $ite(
                                  Xa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_5682_bij__betw__Suc,axiom,
    ! [M7: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M7,N3)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M7) = N3 ) ) ).

% bij_betw_Suc
tff(fact_5683_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),image(nat,A,semiring_1_of_nat(A)),N3) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_5684_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F2: fun(A,fun(B,C))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),A3)
     => aa(set(C),$o,member(C,aa(B,C,aa(A,fun(B,C),F2,A2),B2)),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)),A3)) ) ).

% pair_imageI
tff(fact_5685_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ aa(set(nat),$o,member(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3)) ).

% zero_notin_Suc_image
tff(fact_5686_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Na: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Na))) )
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% nat_seg_image_imp_finite
tff(fact_5687_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),image(nat,A,F7),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),N2))) ) ).

% finite_conv_nat_seg_image
tff(fact_5688_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_lessThan(int),K3)) ) ).

% finite_int_iff_bounded
tff(fact_5689_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_atMost(int),K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_5690_in__image__insert__iff,axiom,
    ! [A: $tType,B3: set(set(A)),X: A,A3: set(A)] :
      ( ! [C8: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),C8),B3)
         => ~ aa(set(A),$o,member(A,X),C8) )
     => ( aa(set(set(A)),$o,member(set(A),A3),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X)),B3))
      <=> ( aa(set(A),$o,member(A,X),A3)
          & aa(set(set(A)),$o,member(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_5691_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_5692_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_5693_Pow__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : pow2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2)),pow2(A,A3))) ).

% Pow_insert
tff(fact_5694_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5695_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5696_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5697_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_5698_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xsa: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xsa))
     => aa(set(set(A)),$o,member(set(A),X5),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa)))) ) ).

% subset_subseqs
tff(fact_5699_subseqs__powset,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa))) = pow2(A,aa(list(A),set(A),set2(A),Xsa)) ).

% subseqs_powset
tff(fact_5700_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_qy(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,minus_minus(int,U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_5701_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),image(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_5702_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_qz(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,X),C2),aa(nat,nat,minus_minus(nat,Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5703_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,X) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),X)
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = zero_zero(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Node(Uu,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_5704_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero(nat),Ts2,S3)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_5705_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),Mi,X),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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))
              & ~ ( ( X = Mi )
                  | ( X = Ma ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_5706_minNull__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(T2)),one_one(nat)) ).

% minNull_bound
tff(fact_5707_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S3)),Xa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)),Xa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                    & ~ ( ( Xa = Mi2 )
                                        | ( Xa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_5708_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
           => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
             => ( ? [N: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xa = Mi2 )
                                & ( Xa = Ma2 ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                        one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_5709_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( ( Xa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B4)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B4: $o] : X = vEBT_Leaf((A4),(B4))
             => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ? [N: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,N))
                 => ( Y != vEBT_Leaf((A4),(B4)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                              vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                              $ite(
                                ( ( Xa = Mi2 )
                                & ( Xa = Ma2 ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                          $let(
                                            newlist: list(vEBT_VEBT),
                                            newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                            $ite(
                                              vEBT_VEBT_minNull(newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                vEBT_Node(some(product_prod(nat,nat),
                                                    aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                        Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                              vEBT_Node(some(product_prod(nat,nat),
                                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                    $ite(xn = Ma2,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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_5710_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_5711_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_5712_VEBT__internal_Osummaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% VEBT_internal.summaxma
tff(fact_5713_VEBT__internal_Onested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,Va2: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( ( Na = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Mi)
         => ( ( Ma != Mi )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)) ) ) ) ) ).

% VEBT_internal.nested_mint
tff(fact_5714_VEBT__internal_Odel__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( 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) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(X = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_not_mi_newnode_not_nil
tff(fact_5715_VEBT__internal_Odel__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_mi_lets_in_not_minNull
tff(fact_5716_VEBT__internal_Odel__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( 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) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    X = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg,Newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(X = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_not_mi
tff(fact_5717_VEBT__internal_Odel__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( 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) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    X = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(X = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_not_mia
tff(fact_5718_VEBT__internal_Odel__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( 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) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    X = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_not_mi_new_node_nil
tff(fact_5719_VEBT__internal_Odel__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_mi_lets_in_minNull
tff(fact_5720_VEBT__internal_Odel__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg,Newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_mi_lets_in
tff(fact_5721_VEBT__internal_Odel__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),X)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                xn: nat,
                xn:= 
                  $ite(X = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(X = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,h),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),minn,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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                  $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary) ) ) ),
                      vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_in_range
tff(fact_5722_VEBT__internal_Odel__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                xn: nat,
                xn:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,h),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary) ) ) ),
                    vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_mia
tff(fact_5723_VEBT__internal_Odel__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( X = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.del_x_mi
tff(fact_5724_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X) ),
        one_one(nat),
        $ite(
          ( ( X = Mi )
          & ( X = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(X = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                    $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                  one_one(nat) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_5725_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X) ),
        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
        $ite(
          ( ( X = Mi )
          & ( X = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
          $let(
            xn: nat,
            xn:= 
              $ite(X = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),
            $let(
              minn: nat,
              minn:= 
                $ite(X = Mi,xn,Mi),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary,h),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                  Ma ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                        vEBT_Node(some(product_prod(nat,nat),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_5726_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_5727_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B4)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B4)) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2)),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                  $ite(
                                    ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                              $let(
                                                newlist: list(vEBT_VEBT),
                                                newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                $ite(
                                                  vEBT_VEBT_minNull(newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    vEBT_Node(some(product_prod(nat,nat),
                                                        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                            Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                                  vEBT_Node(some(product_prod(nat,nat),
                                                      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_5728_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(X,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
           => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
             => ( ? [N: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                                | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                              $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                          one_one(nat))),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(
                                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                  $let(
                                                    newlist: list(vEBT_VEBT),
                                                    newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                      $ite(
                                                        vEBT_VEBT_minNull(newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary2,h),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                  $let(
                                                                    maxs: option(nat),
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                      $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                one_one(nat) )) )),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                          $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                              one_one(nat) ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_5729_maxt__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(T2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_5730_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
        $ite(
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X) ),
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
            $ite(
              ( ( X = Mi )
              & ( X = Ma ) ),
              aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
              aa(nat,nat,
                aa(nat,fun(nat,nat),plus_plus(nat),
                  aa(nat,nat,
                    aa(nat,fun(nat,nat),plus_plus(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                        $ite(X = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                    one_one(nat))),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(X = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $let(
                      h: nat,
                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l),
                            $let(
                              newlist: list(vEBT_VEBT),
                              newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                $ite(
                                  vEBT_VEBT_minNull(newnode),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary,h),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                        $ite(
                                          xn = Ma,
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                            $let(
                                              maxs: option(nat),
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                          one_one(nat) )) )),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                    $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                        one_one(nat) ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_5731_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2)
                                    | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                      $ite(
                                        ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                                  $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                              one_one(nat))),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(
                                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                      $let(
                                                        newlist: list(vEBT_VEBT),
                                                        newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                          $ite(
                                                            vEBT_VEBT_minNull(newnode),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                      $let(
                                                                        maxs: option(nat),
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                          $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                    one_one(nat) )) )),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_5732_vebt__delete__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(Info,Deg,TreeList,Summary),X) = case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(Info,Deg,TreeList,Summary),aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_rb(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Info),Deg),TreeList),Summary),X),Info) ).

% vebt_delete_code(2)
tff(fact_5733_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( ! [Uu: $o,Uv: $o] : X != vEBT_Leaf((Uu),(Uv))
       => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_5734_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_qf(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_5735_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_5736_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_5737_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd),X)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_5738_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_5739_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_5740_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : X = vEBT_Leaf((Uu),(Uv))
         => (Y) )
       => ( ( ? [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_5741_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xa)) ) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Xa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa)) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_5742_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X = vEBT_Leaf((Uu),(Uv)) )
             => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xa)) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Xa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa))
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa))
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_5743_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa))
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xa))
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_5744_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_dc(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_5745_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_rc(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_5746_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),X: option(A)] :
      ( case_option($o,A,(P),Q,X)
     => ( ( ( X = none(A) )
         => ~ (P) )
       => ~ ! [Y3: A] :
              ( ( X = some(A,Y3) )
             => ~ aa(A,$o,Q,Y3) ) ) ) ).

% case_optionE
tff(fact_5747_vebt__insert__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(Info,Deg,TreeList,Summary)),X) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),vEBT_Node(Info,Deg,TreeList,Summary),case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),X)),Deg,TreeList,Summary),aa(fun(nat,fun(nat,vEBT_VEBT)),fun(product_prod(nat,nat),vEBT_VEBT),product_case_prod(nat,nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_rd(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Deg),TreeList),Summary),X)),Info)) ).

% vebt_insert_code(2)
tff(fact_5748_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa = zero_zero(nat),
                    (A4),
                    $ite(Xa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
           => (Y) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_5749_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xa = zero_zero(nat),
                  (A4),
                  $ite(Xa = one_one(nat),(B4),$false) ) )
       => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_5750_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_5751_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_5752_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_5753_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_5754_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_5755_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).

% max_less_iff_conj
tff(fact_5756_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_5757_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_5758_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)) ).

% max_Suc_Suc
tff(fact_5759_max__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),zero_zero(nat)) = Na ).

% max_0R
tff(fact_5760_max__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Na) = Na ).

% max_0L
tff(fact_5761_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_5762_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_5763_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_5764_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_5765_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_5766_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_5767_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_5768_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_5769_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_5770_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_5771_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_5772_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_5773_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_5774_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_5775_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_5776_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_5777_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Na)) ).

% max_numeral_Suc
tff(fact_5778_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ).

% Max_insert
tff(fact_5779_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_5780_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_5781_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_5782_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa3: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3),Xa3,X3) ) ).

% max_def_raw
tff(fact_5783_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_5784_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_5785_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_5786_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_5787_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_5788_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_5789_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_5790_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_5791_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_5792_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_5793_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_5794_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_5795_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_5796_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,minus_minus(A,X),Z2)),aa(A,A,minus_minus(A,Y),Z2)) ) ).

% max_diff_distrib_left
tff(fact_5797_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_5798_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Q3)) ).

% nat_add_max_left
tff(fact_5799_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).

% nat_add_max_right
tff(fact_5800_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).

% max_add_distrib_right
tff(fact_5801_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).

% max_add_distrib_left
tff(fact_5802_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_5803_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_5804_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_5805_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_5806_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_5807_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y)) ) ).

% of_int_max
tff(fact_5808_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_5809_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_max_right
tff(fact_5810_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ).

% nat_mult_max_left
tff(fact_5811_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Na),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M) ).

% nat_minus_add_max
tff(fact_5812_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ).

% Max.in_idem
tff(fact_5813_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_5814_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X4: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X4)),aa(A,A,H,Y3))
         => ( aa(set(A),$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),image(A,A,H),N3)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_5815_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_5816_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,X),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_5817_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y3: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3) ) ) ) ) ).

% Max.closed
tff(fact_5818_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_5819_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),X)
    <=> $ite(
          X = zero_zero(nat),
          (A2),
          $ite(X = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_5820_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_5821_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_5822_VEBT__internal_Oheight__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))) ) ).

% VEBT_internal.height_i_max
tff(fact_5823_VEBT__internal_Omax__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Na: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ) ).

% VEBT_internal.max_idx_list
tff(fact_5824_VEBT__internal_Oinsert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( X != Ma )
           => ( aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Summary),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% VEBT_internal.insert_simp_excp
tff(fact_5825_VEBT__internal_Oinsert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( X != Ma )
           => ( aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Summary),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% VEBT_internal.insert_simp_norm
tff(fact_5826_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),Mi,X),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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))
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) ),
            vEBT_Node(some(product_prod(nat,nat),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),X,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,aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Summary),h),Summary)),
            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_5827_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S),X)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_5828_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,X),Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  Xa = zero_zero(nat),
                  vEBT_Leaf($true,(B4)),
                  $ite(Xa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
              ( ( X = 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] :
                ( ( X = 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) ) )
           => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) ),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Xa,Mi2)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Summary2),h),Summary2)),
                              vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_5829_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xa = zero_zero(nat),
                (A4),
                $ite(Xa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X != vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_5830_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,X),Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      Xa = zero_zero(nat),
                      vEBT_Leaf($true,(B4)),
                      $ite(Xa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S3)),Xa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = 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) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)),Xa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Mi2,Xa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xa = Mi2 )
                                      | ( Xa = Ma2 ) ) ),
                                  vEBT_Node(some(product_prod(nat,nat),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),Xa,Mi2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Summary2),h),Summary2)),
                                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_5831_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xa = zero_zero(nat),
                      (A4),
                      $ite(Xa = one_one(nat),(B4),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Xa)) ) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_5832_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa))
               => ~ $ite(
                      Xa = zero_zero(nat),
                      (A4),
                      $ite(Xa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)),Xa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_5833_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q3),zero_zero(extended_enat)) = Q3 ).

% max_enat_simps(2)
tff(fact_5834_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_5835_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_5836_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa))
               => $ite(
                    Xa = zero_zero(nat),
                    (A4),
                    $ite(Xa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Xa)) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V4),TreeList2,S3)),Xa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_5837_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xa = zero_zero(nat),
                (A4),
                $ite(Xa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
         => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)
           => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => $ite(
                        Xa = Mi2,
                        $true,
                        $ite(
                          Xa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_5838_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(X),Xa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa = zero_zero(nat),
                    (A4),
                    $ite(Xa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => (Y) )
         => ( ( ? [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)
             => (Y) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xa = Mi2,
                            $true,
                            $ite(
                              Xa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_5839_set__vebt__member,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( aa(nat,$o,vEBT_vebt_member(T2),X)
      <=> aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2)) ) ) ).

% set_vebt_member
tff(fact_5840_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_5841_VEBT__internal_Oset__vebt_H__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_VEBT_set_vebt(T2) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(T2)) ).

% VEBT_internal.set_vebt'_def
tff(fact_5842_vebt__member__code_I2_J,axiom,
    ! [T2: nat,R3: list(vEBT_VEBT),E2: vEBT_VEBT,X: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),T2,R3,E2)),X) ).

% vebt_member_code(2)
tff(fact_5843_vebt__inst_Oset__vebt__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_vebt_member(T2),X)
      <=> aa(set(nat),$o,member(nat,X),vEBT_set_vebt(T2)) ) ) ).

% vebt_inst.set_vebt_member
tff(fact_5844_VEBT__internal_Ogreater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
    <=> vEBT_VEBT_greater(some(nat,X),some(nat,Y)) ) ).

% VEBT_internal.greater_shift
tff(fact_5845_vebt__member__code_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),X)
    <=> $ite(
          X = zero_zero(nat),
          (A2),
          $ite(X = one_one(nat),(B2),$false) ) ) ).

% vebt_member_code(1)
tff(fact_5846_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Uy2,Uz)),X) ).

% vebt_member.simps(3)
tff(fact_5847_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),X) ).

% vebt_member.simps(4)
tff(fact_5848_VEBT__internal_Omint__corr__help,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Mini: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = some(nat,Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(T2),X)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),X) ) ) ) ).

% VEBT_internal.mint_corr_help
tff(fact_5849_VEBT__internal_Omaxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Maxi: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = some(nat,Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(T2),X)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Maxi) ) ) ) ).

% VEBT_internal.maxt_corr_help
tff(fact_5850_VEBT__internal_Omember__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,Na: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),X)
     => ( vEBT_invar_vebt(Tree,Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% VEBT_internal.member_bound
tff(fact_5851_VEBT__internal_Opost__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( 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))),Na))
         => ( aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,T2),X)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(T2),Y)
              | ( X = Y ) ) ) ) ) ) ).

% VEBT_internal.post_member_pre_member
tff(fact_5852_VEBT__internal_Omember__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( X = Mi )
          | ( X = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% VEBT_internal.member_inv
tff(fact_5853_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X)
    <=> $ite(
          X = Mi,
          $true,
          $ite(
            X = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_5854_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X)
    <=> $ite(
          ( ( Deg = zero_zero(nat) )
          | ( Deg = aa(nat,nat,suc,zero_zero(nat)) ) ),
          $false,
          $ite(
            X = Mi,
            $true,
            $ite(
              X = Ma,
              $true,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
                $false,
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
                  $false,
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
tff(fact_5855_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xa = zero_zero(nat),
                  (A4),
                  $ite(Xa = one_one(nat),(B4),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
             => ~ $ite(
                    Xa = Mi2,
                    $true,
                    $ite(
                      Xa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_5856_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(X,Xa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : X = vEBT_Leaf((Uu),(Uv))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw: $o] : X = vEBT_Leaf((A4),(Uw))
           => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ? [Va: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                  $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_5857_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(X),Xa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xa = zero_zero(nat),
                      (A4),
                      $ite(Xa = one_one(nat),(B4),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2) )
                 => ( ~ (Y)
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xa = Mi2,
                              $true,
                              $ite(
                                Xa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_5858_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(X),Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa))
               => $ite(
                    Xa = zero_zero(nat),
                    (A4),
                    $ite(Xa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa)) )
           => ( ! [V4: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy,Uz2)),Xa)) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa))
                       => $ite(
                            Xa = Mi2,
                            $true,
                            $ite(
                              Xa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_5859_set__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,X: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_pred(T2,X) = some(nat,Px) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),X,Px) ) ) ).

% set_vebt_pred
tff(fact_5860_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_5861_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vd,Ve),Vf) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_5862_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_5863_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A2),some(nat,zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_5864_vebt__pred__code,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),X) = $ite(
        X = zero_zero(nat),
        none(nat),
        $ite(
          X = one_one(nat),
          $ite((A2),some(nat,zero_zero(nat)),none(nat)),
          $ite(
            (B2),
            some(nat,one_one(nat)),
            $ite((A2),some(nat,zero_zero(nat)),none(nat)) ) ) ) ).

% vebt_pred_code
tff(fact_5865_VEBT__internal_Opred__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,X) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_re(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% VEBT_internal.pred_empty
tff(fact_5866_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va2: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        (B2),
        some(nat,one_one(nat)),
        $ite((A2),some(nat,zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_5867_VEBT__internal_Ohelpypredd,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,X) = some(nat,Y) )
       => 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))),Na)) ) ) ).

% VEBT_internal.helpypredd
tff(fact_5868_VEBT__internal_Opred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X)
       => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = some(nat,Ma) ) ) ) ).

% VEBT_internal.pred_max
tff(fact_5869_VEBT__internal_Opred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% VEBT_internal.pred_list_to_short
tff(fact_5870_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_p_r_e_d2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_5871_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(X,Xa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : X = vEBT_Leaf((Uu),(Uv))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw: $o] : X = vEBT_Leaf((A4),(Uw))
           => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : X = vEBT_Leaf((A4),(B4))
             => ( ? [Va: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_5872_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_p_r_e_d(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                      $ite(
                        ( ( minlow != none(nat) )
                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summary,h),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                            $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_5873_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(X),Xa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xa))
               => ~ $ite(
                      Xa = zero_zero(nat),
                      (A4),
                      $ite(Xa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa))
                 => ~ $ite(
                        Xa = Mi2,
                        $true,
                        $ite(
                          Xa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_5874_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)),Xa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_5875_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)),Xa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_5876_VEBT__internal_Oless__shift,axiom,
    ! [X: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
    <=> vEBT_VEBT_less(some(nat,X),some(nat,Y)) ) ).

% VEBT_internal.less_shift
tff(fact_5877_is__pred__in__set__def,axiom,
    ! [Xsa: set(nat),X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xsa,X,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xsa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
        & ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),Xsa)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),X)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_5878_VEBT__internal_Opred__none__empty,axiom,
    ! [Xsa: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_pred_in_set(Xsa,A2,X_13)
     => ( aa(set(nat),$o,finite_finite2(nat),Xsa)
       => ~ ? [X3: nat] :
              ( aa(set(nat),$o,member(nat,X3),Xsa)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),A2) ) ) ) ).

% VEBT_internal.pred_none_empty
tff(fact_5879_VEBT__internal_Oobtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z2),X)
     => ( vEBT_VEBT_min_in_set(A3,Z2)
       => ( aa(set(nat),$o,finite_finite2(nat),A3)
         => ? [X_13: nat] : vEBT_is_pred_in_set(A3,X,X_13) ) ) ) ).

% VEBT_internal.obtain_set_pred
tff(fact_5880_VEBT__internal_Opred__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T2),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
        & ! [Z6: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T2),Z6)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z6),X) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z6),Y) ) ) ) ).

% VEBT_internal.pred_member
tff(fact_5881_vebt__inst_Oset__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,X) = some(nat,Px) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),X,Px) ) ) ).

% vebt_inst.set_vebt_pred
tff(fact_5882_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(X,Xa) = Y )
     => ( ( ? [Uu: $o,B4: $o] : X = vEBT_Leaf((Uu),(B4))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : X = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_5883_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,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,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summary,h),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_5884_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(X,Xa) = Y )
     => ( ( ? [Uu: $o,B4: $o] : X = vEBT_Leaf((Uu),(B4))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : X = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_5885_set__vebt__succ,axiom,
    ! [T2: vEBT_VEBT,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( ( vEBT_vebt_succ(T2,X) = some(nat,Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% set_vebt_succ
tff(fact_5886_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),V2),zero_zero(nat),Vc,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_5887_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_5888_vebt__succ__code_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      vEBT_vebt_succ(vEBT_Leaf((A2),(B2)),X) = $ite(
        ( (B2)
        & ( X = zero_zero(nat) ) ),
        some(nat,one_one(nat)),
        none(nat) ) ).

% vebt_succ_code(1)
tff(fact_5889_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = $ite((B2),some(nat,one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_5890_VEBT__internal_Osucc__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,X) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_rf(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% VEBT_internal.succ_empty
tff(fact_5891_VEBT__internal_OgeqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),X)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ).

% VEBT_internal.geqmaxNone
tff(fact_5892_VEBT__internal_Ohelpyd,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,X) = some(nat,Y) )
       => 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))),Na)) ) ) ).

% VEBT_internal.helpyd
tff(fact_5893_VEBT__internal_Osucc__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = some(nat,Mi) ) ) ) ).

% VEBT_internal.succ_min
tff(fact_5894_VEBT__internal_Osucc__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),X)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% VEBT_internal.succ_list_to_short
tff(fact_5895_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_5896_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( X = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                        $ite(
                                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_5897_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( X = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_5898_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : X = vEBT_Leaf((Uu),(Uv))
         => ( ( Xa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw: $o] : X = vEBT_Leaf((A4),(Uw))
             => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A4),some(nat,zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X = vEBT_Leaf((A4),(B4)) )
               => ( ? [Va: nat] : Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( Y != $ite(
                        (B4),
                        some(nat,one_one(nat)),
                        $ite((A4),some(nat,zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                              some(nat,Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                        vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xa),some(nat,Mi2),none(nat)),
                                            vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),pr),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_5899_is__succ__in__set__def,axiom,
    ! [Xsa: set(nat),X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xsa,X,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xsa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
        & ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),Xsa)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),X2)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X2) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_5900_VEBT__internal_Osucc__none__empty,axiom,
    ! [Xsa: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_succ_in_set(Xsa,A2,X_13)
     => ( aa(set(nat),$o,finite_finite2(nat),Xsa)
       => ~ ? [X3: nat] :
              ( aa(set(nat),$o,member(nat,X3),Xsa)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X3) ) ) ) ).

% VEBT_internal.succ_none_empty
tff(fact_5901_VEBT__internal_Oobtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A3: set(nat),B3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Z2)
     => ( vEBT_VEBT_max_in_set(A3,Z2)
       => ( aa(set(nat),$o,finite_finite2(nat),B3)
         => ( ( A3 = B3 )
           => ? [X_13: nat] : vEBT_is_succ_in_set(A3,X,X_13) ) ) ) ) ).

% VEBT_internal.obtain_set_succ
tff(fact_5902_VEBT__internal_Osucc__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T2),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
        & ! [Z6: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T2),Z6)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Z6) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z6) ) ) ) ).

% VEBT_internal.succ_member
tff(fact_5903_vebt__inst_Oset__vebt__succ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,X) = some(nat,Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% vebt_inst.set_vebt_succ
tff(fact_5904_vebt__succ__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_vebt_succ(vEBT_Node(Info,Deg,TreeList,Summary),X) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),none(nat),case_option(option(nat),product_prod(nat,nat),none(nat),aa(fun(nat,fun(nat,option(nat))),fun(product_prod(nat,nat),option(nat)),product_case_prod(nat,nat,option(nat)),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_rg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Deg),TreeList),Summary),X)),Info)) ).

% vebt_succ_code(2)
tff(fact_5905_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Mi),
        some(nat,Mi),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summary,h),
                    $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_5906_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),X) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),X),
        some(nat,Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                  vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summary,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X),some(nat,Mi),none(nat)),
                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),pr),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_5907_VEBT__internal_Osucc__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),X)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                l: nat,
                l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.succ_less_length_list
tff(fact_5908_VEBT__internal_Osucc__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),X)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
              l: nat,
              l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% VEBT_internal.succ_greatereq_min
tff(fact_5909_VEBT__internal_Opred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
                l: nat,
                l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(some(nat,l),minlow) ),
                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X),some(nat,Mi),none(nat)),
                          vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),pr),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.pred_less_length_list
tff(fact_5910_VEBT__internal_Opred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = $let(
              l: nat,
              l:= vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(some(nat,l),minlow) ),
                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X),some(nat,Mi),none(nat)),
                          vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),pr),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% VEBT_internal.pred_lesseq_max
tff(fact_5911_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa) = Y )
     => ( ! [Uu: $o,B4: $o] :
            ( ( X = vEBT_Leaf((Uu),(B4)) )
           => ( ( Xa = zero_zero(nat) )
             => ( Y != $ite((B4),some(nat,one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : X = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                            some(nat,Mi2),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                      vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_5912_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( X = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = $ite((B4),some(nat,one_one(nat)),none(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi2),
                                some(nat,Mi2),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                          vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_5913_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( X = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A4),some(nat,zero_zero(nat)),none(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = $ite(
                              (B4),
                              some(nat,one_one(nat)),
                              $ite((A4),some(nat,zero_zero(nat)),none(nat)) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve2)),Xa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = none(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xa),
                                  some(nat,Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                            vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xa),some(nat,Mi2),none(nat)),
                                                vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),pr),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_5914_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Na: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = case_option(int,num,zero_zero(int),aTP_Lamp_rh(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5915_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_5916_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa(nat,nat,suc,Na),one2) = some(num,one2) ).

% take_bit_num_simps(2)
tff(fact_5917_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),one2) = some(num,one2) ).

% take_bit_num_simps(5)
tff(fact_5918_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Na),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Na,M)) ).

% take_bit_num_simps(3)
tff(fact_5919_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Na),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Na,M))) ).

% take_bit_num_simps(4)
tff(fact_5920_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(pred_numeral(R3),M)) ).

% take_bit_num_simps(6)
tff(fact_5921_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R3),M))) ).

% take_bit_num_simps(7)
tff(fact_5922_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),Na)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ) ).

% take_bit_numeral_numeral
tff(fact_5923_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: num,Q3: num] :
          ( ( bit_take_bit_num(M,Na) = some(num,Q3) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5924_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: num] :
          ( ( bit_take_bit_num(M,Na) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),Na)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5925_and__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(3)
tff(fact_5926_and__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(7)
tff(fact_5927_and__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Na))) ).

% and_minus_numerals(8)
tff(fact_5928_and__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Na))) ).

% and_minus_numerals(4)
tff(fact_5929_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5930_and__not__num_Osimps_I2_J,axiom,
    ! [Na: num] : bit_and_not_num(one2,bit0(Na)) = some(num,one2) ).

% and_not_num.simps(2)
tff(fact_5931_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one2) = some(num,bit0(M)) ).

% and_not_num.simps(4)
tff(fact_5932_and__not__num_Osimps_I3_J,axiom,
    ! [Na: num] : bit_and_not_num(one2,aa(num,num,bit1,Na)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5933_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = some(num,bit0(M)) ).

% and_not_num.simps(7)
tff(fact_5934_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Na: num,Q3: num] :
      ( ( bit_and_not_num(M,Na) = some(num,Q3) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = aa(num,int,numeral_numeral(int),Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_5935_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(Na)) = case_option(option(num),num,some(num,one2),aTP_Lamp_rj(num,option(num)),bit_and_not_num(M,Na)) ).

% and_not_num.simps(8)
tff(fact_5936_and__not__num__eq__None__iff,axiom,
    ! [M: num,Na: num] :
      ( ( bit_and_not_num(M,Na) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_5937_int__numeral__not__and__num,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Na,M)) ).

% int_numeral_not_and_num
tff(fact_5938_int__numeral__and__not__num,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,Na)) ).

% int_numeral_and_not_num
tff(fact_5939_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),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_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% or_int_unfold
tff(fact_5940_test__def,axiom,
    ! [Na: nat,Xsa: list(nat),Ysa: list(nat)] :
      vEBT_Intf_test(Na,Xsa,Ysa) = $let(
        t: vEBT_VEBT,
        t:= foldl(vEBT_VEBT,nat,vEBT_vebt_insert,vEBT_vebt_buildup(Na),aa(list(nat),list(nat),cons(nat,zero_zero(nat)),Xsa)),
        aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_rk(vEBT_VEBT,fun(nat,nat),t)),Ysa) ) ).

% test_def
tff(fact_5941_set__vebt__pred_H,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_vebt_pred(T2,X) = $ite(
            ? [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),X) ),
            some(nat,order_Greatest(nat,aa(nat,fun(nat,$o),aTP_Lamp_rl(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X))),
            none(nat) ) ) ) ).

% set_vebt_pred'
tff(fact_5942_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),cons(A,X21),X222) = aa(list(A),list(A),cons(A,Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_5943_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_5944_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_5945_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_5946_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_5947_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_5948_bex__empty,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ~ ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),bot_bot(set(A)))
          & aa(A,$o,P,X3) ) ).

% bex_empty
tff(fact_5949_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B2)) ) ).

% take_bit_or
tff(fact_5950_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4730199178511100633sh_bit(A,Na,A2)),bit_se4730199178511100633sh_bit(A,Na,B2)) ) ).

% push_bit_or
tff(fact_5951_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),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_rm(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),Q)))
      <=> ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_rn(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),X2))) ) ) ) ).

% finite_Collect_bex
tff(fact_5952_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xsa),Na) ).

% nth_Cons_Suc
tff(fact_5953_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_5954_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_5955_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_5956_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_5957_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_5958_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_5959_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_5960_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_5961_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_5962_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_5963_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_5964_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_5965_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = 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),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_5966_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_5967_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_5968_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xsa: list(A),V2: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_5969_or__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na))) ).

% or_minus_numerals(6)
tff(fact_5970_or__minus__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na))) ).

% or_minus_numerals(2)
tff(fact_5971_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,X: A,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5972_or__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_5973_and__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_5974_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_5975_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_5976_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,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),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_5977_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list(nat),Q3: nat] :
      ( ( aa(list(nat),list(nat),cons(nat,M),aa(list(nat),list(nat),cons(nat,Na),Ns)) = upt(M,Q3) )
    <=> ( aa(list(nat),list(nat),cons(nat,Na),Ns) = upt(aa(nat,nat,suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_5978_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xsa),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),cons(A,X),list_update(A,Xsa,I,Y)) ).

% list_update_code(3)
tff(fact_5979_Suc__length__conv,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( ( aa(nat,nat,suc,Na) = aa(list(A),nat,size_size(list(A)),Xsa) )
    <=> ? [Y5: A,Ys2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,Y5),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5980_length__Suc__conv,axiom,
    ! [A: $tType,Xsa: list(A),Na: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,suc,Na) )
    <=> ? [Y5: A,Ys2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,Y5),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5981_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_5982_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_5983_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_5984_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xsa: list(A)] :
      remove1(A,X,aa(list(A),list(A),cons(A,Y),Xsa)) = $ite(X = Y,Xsa,aa(list(A),list(A),cons(A,Y),remove1(A,X,Xsa))) ).

% remove1.simps(2)
tff(fact_5985_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_5986_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z2)) ) ).

% bit.conj_disj_distrib
tff(fact_5987_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Z2)) ) ).

% bit.disj_conj_distrib
tff(fact_5988_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z2),X)) ) ).

% bit.conj_disj_distrib2
tff(fact_5989_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z2),X)) ) ).

% bit.disj_conj_distrib2
tff(fact_5990_foldl__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A2: A,X: B,Xsa: list(B)] : foldl(A,B,F2,A2,aa(list(B),list(B),cons(B,X),Xsa)) = foldl(A,B,F2,aa(B,A,aa(A,fun(B,A),F2,A2),X),Xsa) ).

% foldl_Cons
tff(fact_5991_not__Cons__self2,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(list(A),list(A),cons(A,X),Xsa) != Xsa ).

% not_Cons_self2
tff(fact_5992_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_5993_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_5994_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_5995_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),Na)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
            | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Na) ) ) ) ).

% bit_or_iff
tff(fact_5996_bit__or__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),Na)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
        | aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Na) ) ) ).

% bit_or_int_iff
tff(fact_5997_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_5998_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_5999_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_or_eq
tff(fact_6000_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ysa: list(A),Z2: A] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
     => aa(set(list(A)),$o,member(list(A),aa(list(A),list(A),cons(A,Z2),Zs)),shuffles(A,Xsa,aa(list(A),list(A),cons(A,Z2),Ysa))) ) ).

% Cons_in_shuffles_rightI
tff(fact_6001_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ysa: list(A),Z2: A] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
     => aa(set(list(A)),$o,member(list(A),aa(list(A),list(A),cons(A,Z2),Zs)),shuffles(A,aa(list(A),list(A),cons(A,Z2),Xsa),Ysa)) ) ).

% Cons_in_shuffles_leftI
tff(fact_6002_distinct__length__2__or__more,axiom,
    ! [A: $tType,A2: A,B2: A,Xsa: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,A2),aa(list(A),list(A),cons(A,B2),Xsa)))
    <=> ( ( A2 != B2 )
        & distinct(A,aa(list(A),list(A),cons(A,A2),Xsa))
        & distinct(A,aa(list(A),list(A),cons(A,B2),Xsa)) ) ) ).

% distinct_length_2_or_more
tff(fact_6003_foldl__cong,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(A,fun(B,A)),G: fun(A,fun(B,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X4: B] :
              ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),L))
             => ( aa(B,A,aa(A,fun(B,A),F2,A4),X4) = aa(B,A,aa(A,fun(B,A),G,A4),X4) ) )
         => ( foldl(A,B,F2,A2,L) = foldl(A,B,G,B2,K) ) ) ) ) ).

% foldl_cong
tff(fact_6004_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),X222))
     => aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ) ).

% list.set_intros(2)
tff(fact_6005_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(set(A),$o,member(A,X21),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ).

% list.set_intros(1)
tff(fact_6006_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,A2: list(A)] :
      ( aa(set(A),$o,member(A,E2),aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),cons(A,E2),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),cons(A,Z12),Z23) )
             => ~ aa(set(A),$o,member(A,E2),aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6007_set__ConsD,axiom,
    ! [A: $tType,Y: A,X: A,Xsa: list(A)] :
      ( aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xsa)))
     => ( ( Y = X )
        | aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% set_ConsD
tff(fact_6008_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),cons(B,X21),X222)) = aa(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_6009_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,X: A,Xsa: list(A),F2: fun(B,A),Ysa: list(B)] :
      ( ( aa(list(A),list(A),cons(A,X),Xsa) = aa(list(B),list(A),map(B,A,F2),Ysa) )
     => ? [Z: B,Zs2: list(B)] :
          ( ( Ysa = aa(list(B),list(B),cons(B,Z),Zs2) )
          & ( X = aa(B,A,F2,Z) )
          & ( Xsa = aa(list(B),list(A),map(B,A,F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_6010_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xsa: list(B),Y: A,Ysa: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xsa) = aa(list(A),list(A),cons(A,Y),Ysa) )
     => ? [Z: B,Zs2: list(B)] :
          ( ( Xsa = aa(list(B),list(B),cons(B,Z),Zs2) )
          & ( aa(B,A,F2,Z) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs2) = Ysa ) ) ) ).

% map_eq_Cons_D
tff(fact_6011_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,X: A,Xsa: list(A),F2: fun(B,A),Ysa: list(B)] :
      ( ( aa(list(A),list(A),cons(A,X),Xsa) = aa(list(B),list(A),map(B,A,F2),Ysa) )
    <=> ? [Z6: B,Zs3: list(B)] :
          ( ( Ysa = aa(list(B),list(B),cons(B,Z6),Zs3) )
          & ( X = aa(B,A,F2,Z6) )
          & ( Xsa = aa(list(B),list(A),map(B,A,F2),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_6012_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xsa: list(B),Y: A,Ysa: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xsa) = aa(list(A),list(A),cons(A,Y),Ysa) )
    <=> ? [Z6: B,Zs3: list(B)] :
          ( ( Xsa = aa(list(B),list(B),cons(B,Z6),Zs3) )
          & ( aa(B,A,F2,Z6) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs3) = Ysa ) ) ) ).

% map_eq_Cons_conv
tff(fact_6013_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_6014_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( 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_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) ) ) ).

% OR_lower
tff(fact_6015_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_6016_impossible__Cons,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ysa))
     => ( Xsa != aa(list(A),list(A),cons(A,X),Ysa) ) ) ).

% impossible_Cons
tff(fact_6017_set__subset__Cons,axiom,
    ! [A: $tType,Xsa: list(A),X: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xsa))) ).

% set_subset_Cons
tff(fact_6018_image__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(B),set(A),image(B,A,F2),A3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ro(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) ).

% image_def
tff(fact_6019_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xsa: list(A),Ysa: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,Y)),shuffles(A,Xsa,Ysa))),shuffles(A,Xsa,aa(list(A),list(A),cons(A,Y),Ysa))) ).

% Cons_shuffles_subset2
tff(fact_6020_Cons__shuffles__subset1,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Ysa: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,X)),shuffles(A,Xsa,Ysa))),shuffles(A,aa(list(A),list(A),cons(A,X),Xsa),Ysa)) ).

% Cons_shuffles_subset1
tff(fact_6021_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,X),aa(list(A),list(A),cons(A,Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Y),Zs)) ) ) ) ).

% sorted2
tff(fact_6022_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_6023_replicate__Suc,axiom,
    ! [A: $tType,Na: nat,X: A] : replicate(A,aa(nat,nat,suc,Na),X) = aa(list(A),list(A),cons(A,X),replicate(A,Na,X)) ).

% replicate_Suc
tff(fact_6024_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xsa),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xsa) ).

% list_update_code(2)
tff(fact_6025_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,X),Xsa))
    <=> ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
        & distinct(A,Xsa) ) ) ).

% distinct.simps(2)
tff(fact_6026_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A,Ysa: list(A)] : shuffles(A,aa(list(A),list(A),cons(A,X),Xsa),aa(list(A),list(A),cons(A,Y),Ysa)) = 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)),image(list(A),list(A),cons(A,X)),shuffles(A,Xsa,aa(list(A),list(A),cons(A,Y),Ysa)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,Y)),shuffles(A,aa(list(A),list(A),cons(A,X),Xsa),Ysa))) ).

% shuffles.simps(3)
tff(fact_6027_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A2: A,F2: fun(C,B),Xsa: list(C)] : foldl(A,B,G,A2,aa(list(C),list(B),map(C,B,F2),Xsa)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_rp(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F2),A2,Xsa) ).

% foldl_map
tff(fact_6028_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xsa: list(A)] :
      find(A,P,aa(list(A),list(A),cons(A,X),Xsa)) = $ite(aa(A,$o,P,X),some(A,X),find(A,P,Xsa)) ).

% find.simps(2)
tff(fact_6029_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ysa: list(A),Xsa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),aa(list(A),list(A),cons(A,Y),Ysa)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa)))
     => aa(set(list(A)),$o,member(list(A),Ysa),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xsa))) ) ).

% Cons_in_subseqsD
tff(fact_6030_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_6031_Suc__le__length__iff,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xsa))
    <=> ? [X2: A,Ys2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,X2),Ys2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Ys2)) ) ) ).

% Suc_le_length_iff
tff(fact_6032_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y)) ) ).

% bit.xor_def
tff(fact_6033_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_6034_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ysa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,X),Ysa))
        <=> ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Ysa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) )
            & sorted_wrt(A,ord_less_eq(A),Ysa) ) ) ) ).

% sorted_simps(2)
tff(fact_6035_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ysa: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),cons(A,X),Ysa))
        <=> ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Ysa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X2) )
            & sorted_wrt(A,ord_less(A),Ysa) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_6036_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),bit_se4730199178511100633sh_bit(A,Na,one_one(A))) ) ).

% set_bit_eq_or
tff(fact_6037_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6038_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_6039_concat__bit__def,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),bit_se4730199178511100633sh_bit(int,Na,L)) ).

% concat_bit_def
tff(fact_6040_set__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,Na,one_one(int))) ).

% set_bit_int_def
tff(fact_6041_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,X),Xsa)),Y) = $ite(X = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xsa),Y)),one_one(nat)),aa(A,nat,count_list(A,Xsa),Y)) ).

% count_list.simps(2)
tff(fact_6042_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(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_or_iff
tff(fact_6043_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_6044_or__not__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na))) ).

% or_not_numerals(2)
tff(fact_6045_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_6046_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),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_6047_nth__Cons_H,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = $ite(Na = zero_zero(nat),X,aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% nth_Cons'
tff(fact_6048_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xsa: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,X),Xsa) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xsa ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6049_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,X),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),aa(list(A),nat,size_list(A,X),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6050_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_6051_or__not__numerals_I3_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na))) ).

% or_not_numerals(3)
tff(fact_6052_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Na: nat] : aa(list(nat),list(A),map(nat,A,F2),upt(zero_zero(nat),aa(nat,nat,suc,Na))) = aa(list(A),list(A),cons(A,aa(nat,A,F2,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_rq(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),Na))) ).

% map_upt_Suc
tff(fact_6053_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_6054_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xsa: list(A),Na: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = Y )
      <=> ( ( aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6055_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = X )
        <=> ( Na = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6056_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,X),Xsa) = replicate(A,Na,Y) )
    <=> ( ( X = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
        & ( Xsa = replicate(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_6057_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_6058_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),bit_se2239418461657761734s_mask(A,Na)) ) ).

% mask_Suc_exp
tff(fact_6059_or__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = 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_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% or_not_numerals(6)
tff(fact_6060_Pow__set_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xsa))) = $let(
        a2: set(set(A)),
        a2:= pow2(A,aa(list(A),set(A),set2(A),Xsa)),
        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)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X)),a2)) ) ).

% Pow_set(2)
tff(fact_6061_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_6062_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_6063_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Na))) ) ).

% mask_Suc_double
tff(fact_6064_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_6065_or__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(5)
tff(fact_6066_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)))) ) ).

% signed_take_bit_def
tff(fact_6067_or__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(9)
tff(fact_6068_or__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(8)
tff(fact_6069_vebt__inst_Oset__vebt__pred_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_pred(T2,X) = $ite(
            ? [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),X) ),
            some(nat,order_Greatest(nat,aa(nat,fun(nat,$o),aTP_Lamp_rl(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X))),
            none(nat) ) ) ) ).

% vebt_inst.set_vebt_pred'
tff(fact_6070_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($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_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_6071_map__project__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A3: set(B)] : map_project(B,A,F2,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_rr(fun(B,option(A)),fun(set(B),fun(A,$o)),F2),A3)) ).

% map_project_def
tff(fact_6072_set__vebt__succ_H,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( vEBT_invar_vebt(T2,n)
     => ( vEBT_vebt_succ(T2,X) = $ite(
            ? [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),X2) ),
            some(nat,ord_Least(nat,aa(nat,fun(nat,$o),aTP_Lamp_rs(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X))),
            none(nat) ) ) ) ).

% set_vebt_succ'
tff(fact_6073_or__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Na)))) ).

% or_minus_numerals(5)
tff(fact_6074_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_6075_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_6076_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_6077_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),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_6078_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_6079_or__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(8)
tff(fact_6080_or__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(4)
tff(fact_6081_or__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(3)
tff(fact_6082_or__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(7)
tff(fact_6083_or__minus__numerals_I1_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Na)))) ).

% or_minus_numerals(1)
tff(fact_6084_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% set_bit_nat_def
tff(fact_6085_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_6086_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_6087_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_6088_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y3: A] :
                  ( aa(A,$o,P,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),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 = X3 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_6089_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z2: A] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y3: A] :
                  ( aa(A,$o,P,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),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 = X3 ) ) )
         => ( aa(A,$o,P,Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z2) ) ) ) ).

% Least1_le
tff(fact_6090_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_6091_Least__equality,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) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_6092_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_6093_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
         => ( ! [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_6094_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_6095_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
         => ( ! [X4: A] :
                ( aa(A,$o,P,X4)
               => aa(A,$o,Q,X4) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_ex
tff(fact_6096_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI_ex
tff(fact_6097_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A2)
         => ( ! [X4: A] :
                ( aa(A,$o,P,X4)
               => aa(A,$o,Q,X4) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2
tff(fact_6098_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Na: nat,Q: fun(nat,$o),M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,Q,M)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q,K2) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_6099_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(bit0(Na),one2) = bit0(one2) ).

% or_not_num_neg.simps(4)
tff(fact_6100_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),aa(num,num,bit1,M)) = bit0(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(6)
tff(fact_6101_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_6102_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_6103_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(5)
tff(fact_6104_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(9)
tff(fact_6105_Least__Suc,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_rt(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_6106_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% or_nat_def
tff(fact_6107_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( ? [X_1: A] : aa(A,$o,P,X_1)
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_6108_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit0(M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_6109_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(8)
tff(fact_6110_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( X = one2 )
           => ! [M4: num] :
                ( ( Xa = bit0(M4) )
               => ( Y != aa(num,num,bit1,M4) ) ) )
         => ( ( ( X = one2 )
             => ! [M4: num] :
                  ( ( Xa = aa(num,num,bit1,M4) )
                 => ( Y != aa(num,num,bit1,M4) ) ) )
           => ( ( ? [N: num] : X = bit0(N)
               => ( ( Xa = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N: num] :
                    ( ( X = bit0(N) )
                   => ! [M4: num] :
                        ( ( Xa = bit0(M4) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
               => ( ! [N: num] :
                      ( ( X = bit0(N) )
                     => ! [M4: num] :
                          ( ( Xa = aa(num,num,bit1,M4) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M4)) ) ) )
                 => ( ( ? [N: num] : X = aa(num,num,bit1,N)
                     => ( ( Xa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa(num,num,bit1,N) )
                         => ! [M4: num] :
                              ( ( Xa = bit0(M4) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = aa(num,num,bit1,N) )
                           => ! [M4: num] :
                                ( ( Xa = aa(num,num,bit1,M4) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_6111_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Na))) ).

% int_numeral_or_not_num_neg
tff(fact_6112_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Na,M))) ).

% int_numeral_not_or_num_neg
tff(fact_6113_numeral__or__not__num__eq,axiom,
    ! [M: num,Na: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Na)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% numeral_or_not_num_eq
tff(fact_6114_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na))) ).

% or_Suc_0_eq
tff(fact_6115_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na))) ).

% Suc_0_or_eq
tff(fact_6116_or__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = 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)),M)
            | ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na) ))),
        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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_6117_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_6118_vebt__inst_Oset__vebt__succ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,X: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_succ(T2,X) = $ite(
            ? [X2: nat] :
              ( aa(set(nat),$o,member(nat,X2),vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),X2) ),
            some(nat,ord_Least(nat,aa(nat,fun(nat,$o),aTP_Lamp_rs(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),X))),
            none(nat) ) ) ) ).

% vebt_inst.set_vebt_succ'
tff(fact_6119_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = one2 )
               => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [M4: num] :
                  ( ( Xa = bit0(M4) )
                 => ( ( Y = aa(num,num,bit1,M4) )
                   => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(M4))) ) ) )
           => ( ( ( X = one2 )
               => ! [M4: num] :
                    ( ( Xa = aa(num,num,bit1,M4) )
                   => ( ( Y = aa(num,num,bit1,M4) )
                     => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M4))) ) ) )
             => ( ! [N: num] :
                    ( ( X = bit0(N) )
                   => ( ( Xa = one2 )
                     => ( ( Y = bit0(one2) )
                       => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N)),one2)) ) ) )
               => ( ! [N: num] :
                      ( ( X = bit0(N) )
                     => ! [M4: num] :
                          ( ( Xa = bit0(M4) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                           => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N)),bit0(M4))) ) ) )
                 => ( ! [N: num] :
                        ( ( X = bit0(N) )
                       => ! [M4: num] :
                            ( ( Xa = aa(num,num,bit1,M4) )
                           => ( ( Y = bit0(bit_or_not_num_neg(N,M4)) )
                             => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N)),aa(num,num,bit1,M4))) ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa(num,num,bit1,N) )
                         => ( ( Xa = one2 )
                           => ( ( Y = one2 )
                             => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),one2)) ) ) )
                     => ( ! [N: num] :
                            ( ( X = aa(num,num,bit1,N) )
                           => ! [M4: num] :
                                ( ( Xa = bit0(M4) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                                 => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),bit0(M4))) ) ) )
                       => ~ ! [N: num] :
                              ( ( X = aa(num,num,bit1,N) )
                             => ! [M4: num] :
                                  ( ( Xa = aa(num,num,bit1,M4) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                                   => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N)),aa(num,num,bit1,M4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_6120_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,$o),aTP_Lamp_ru(set(A),fun(fun(A,B),fun(A,$o)),S2),F2)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_6121_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)) )
           => aa(set(A),$o,member(A,lattic7623131987881927897min_on(A,B,F2,S2)),S2) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_6122_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)) )
           => ( aa(set(A),$o,member(A,Y),S2)
             => aa(B,$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_6123_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)) )
           => ~ ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),S2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S2))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_6124_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_rv(num,fun(nat,option(num)),M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_6125_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_6126_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_6127_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V2: num,Na: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Na)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),Na)) ).

% case_nat_add_eq_if
tff(fact_6128_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_rw(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_6129_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_6130_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_6131_nth__Cons,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = case_nat(A,X,nth(A,Xsa),Na) ).

% nth_Cons
tff(fact_6132_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,X: A,F2: fun(nat,A),Na: nat] :
      case_nat(A,X,F2,Na) = $ite(Na = zero_zero(nat),X,aa(nat,A,F2,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_6133_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Na: nat] : bit_take_bit_num(Na,one2) = case_nat(option(num),none(num),aTP_Lamp_rx(nat,option(num)),Na) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_6134_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,bit0(M)) = case_nat(option(num),none(num),aTP_Lamp_ry(num,fun(nat,option(num)),M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_6135_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_sc(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),Na),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_6136_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),X: option(A),R2: fun(B,$o),F2: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,X)
     => ( ( (P)
         => aa(B,$o,R2,F2) )
       => ( ! [Q5: A] :
              ( aa(A,$o,Q,Q5)
             => aa(B,$o,R2,aa(A,B,G,Q5)) )
         => aa(B,$o,R2,case_option(B,A,F2,G,X)) ) ) ) ).

% disjE_realizer2
tff(fact_6137_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_sd(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_6138_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_se(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_6139_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X22: num] : case_num(A,F1,F22,F32,bit0(X22)) = aa(num,A,F22,X22) ).

% verit_eq_simplify(17)
tff(fact_6140_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F22,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_6141_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F22,F32,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_sf(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_sf(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32),Num) ).

% num.case_distrib
tff(fact_6142_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : case_num(A,F1,F22,F32,aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_6143_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_6144_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_sg(nat,fun(nat,nat),Na),M) ).

% max_Suc2
tff(fact_6145_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),M) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_sh(nat,fun(nat,nat),Na),M) ).

% max_Suc1
tff(fact_6146_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),I: nat,V2: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xsa),I,V2) = case_nat(list(A),aa(list(A),list(A),cons(A,V2),Xsa),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_si(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xsa),V2),I) ).

% list_update.simps(2)
tff(fact_6147_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ig(nat,nat),aa(nat,nat,minus_minus(nat,M),Na)) ).

% diff_Suc
tff(fact_6148_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W2))),Na)
        <=> case_nat($o,$false,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),Na) ) ) ).

% bit_numeral_rec(1)
tff(fact_6149_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W2))),Na)
        <=> case_nat($o,$true,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),Na) ) ) ).

% bit_numeral_rec(2)
tff(fact_6150_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_sj(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A3) ) ) ).

% Pow_fold
tff(fact_6151_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A,Ysa: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xsa)),aa(list(A),list(A),cons(A,Y),Ysa)))
     => ( shuffles(A,aa(list(A),list(A),cons(A,X),Xsa),aa(list(A),list(A),cons(A,Y),Ysa)) = 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)),image(list(A),list(A),cons(A,X)),shuffles(A,Xsa,aa(list(A),list(A),cons(A,Y),Ysa)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,Y)),shuffles(A,aa(list(A),list(A),cons(A,X),Xsa),Ysa))) ) ) ).

% shuffles.psimps(3)
tff(fact_6152_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),insert(A),B3,A3) ) ) ).

% union_fold_insert
tff(fact_6153_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,ord_max(A),X,A3) ) ) ) ).

% Max.eq_fold
tff(fact_6154_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),image(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_sk(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_6155_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa),bot_bot(set(list(A)))) )
             => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ( ( ( Xa = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) )
               => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))) ) )
           => ~ ! [X4: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),cons(A,X4),Xs2) )
                 => ! [Y3: A,Ys3: list(A)] :
                      ( ( Xa = aa(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)),image(list(A),list(A),cons(A,X4)),shuffles(A,Xs2,aa(list(A),list(A),cons(A,Y3),Ys3)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,Y3)),shuffles(A,aa(list(A),list(A),cons(A,X4),Xs2),Ys3))) )
                       => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,Y3),Ys3))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_6156_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat),none(num),some(num,aa(nat,num,num_of_nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6157_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_6158_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F2),Xsa) )
    <=> ( Xsa = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_6159_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xsa) = nil(A) )
    <=> ( Xsa = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_6160_list__update__nonempty,axiom,
    ! [A: $tType,Xsa: list(A),K: nat,X: A] :
      ( ( list_update(A,Xsa,K,X) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% list_update_nonempty
tff(fact_6161_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6162_Nil__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),nil(A)),shuffles(A,Xsa,Ysa))
    <=> ( ( Xsa = nil(A) )
        & ( Ysa = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_6163_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : aa(nat,num,num_of_nat,aa(num,nat,numeral_numeral(nat),Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_6164_length__0__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = zero_zero(nat) )
    <=> ( Xsa = nil(A) ) ) ).

% length_0_conv
tff(fact_6165_set__empty,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xsa) = bot_bot(set(A)) )
    <=> ( Xsa = nil(A) ) ) ).

% set_empty
tff(fact_6166_set__empty2,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xsa) )
    <=> ( Xsa = nil(A) ) ) ).

% set_empty2
tff(fact_6167_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_6168_replicate__empty,axiom,
    ! [A: $tType,Na: nat,X: A] :
      ( ( replicate(A,Na,X) = nil(A) )
    <=> ( Na = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_6169_empty__replicate,axiom,
    ! [A: $tType,Na: nat,X: A] :
      ( ( nil(A) = replicate(A,Na,X) )
    <=> ( Na = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_6170_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_6171_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_6172_n__lists__Nil,axiom,
    ! [A: $tType,Na: nat] :
      n_lists(A,Na,nil(A)) = $ite(Na = zero_zero(nat),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_6173_length__greater__0__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xsa))
    <=> ( Xsa != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_6174_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)),aa(list(nat),list(nat),cons(nat,aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na))),nil(nat)) ).

% upt_rec_numeral
tff(fact_6175_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F4: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F4: fun(A,B),A4: A,As: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_6176_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ( ! [P5: fun(A,fun(A,$o)),X4: A] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),nil(A)))
       => ~ ! [P5: fun(A,fun(A,$o)),X4: A,Y3: A,Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2))) ) ) ).

% successively.cases
tff(fact_6177_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F4: fun(A,B),X4: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F4),aa(list(A),list(A),cons(A,X4),nil(A)))
         => ( ! [F4: fun(A,B),X4: A,Y3: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F4),aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Zs2)))
           => ~ ! [A4: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_6178_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ~ ! [P5: fun(A,fun(A,$o)),X4: A,Ys3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_6179_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))
       => ~ ! [X4: A,Xs2: list(A),Y3: A,Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,Y3),Ys3)) ) ) ).

% shuffles.cases
tff(fact_6180_splice_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ~ ! [X4: A,Xs2: list(A),Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),Ys3) ) ).

% splice.cases
tff(fact_6181_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),cons(A,X21),X222) ).

% list.distinct(1)
tff(fact_6182_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),cons(A,X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_6183_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),cons(A,X212),X223) ) ).

% list.exhaust
tff(fact_6184_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A)] :
          ( ! [X4: A,Xs2: list(A)] : X != aa(list(A),list(A),cons(A,X4),Xs2)
         => ( X = nil(A) ) ) ) ).

% min_list.cases
tff(fact_6185_foldl__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),A2: A] : foldl(A,B,F2,A2,nil(B)) = A2 ).

% foldl_Nil
tff(fact_6186_transpose_Ocases,axiom,
    ! [A: $tType,X: list(list(A))] :
      ( ( X != nil(list(A)) )
     => ( ! [Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss)
       => ~ ! [X4: A,Xs2: list(A),Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss) ) ) ).

% transpose.cases
tff(fact_6187_remdups__adj_Ocases,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( X != nil(A) )
     => ( ! [X4: A] : X != aa(list(A),list(A),cons(A,X4),nil(A))
       => ~ ! [X4: A,Y3: A,Xs2: list(A)] : X != aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_6188_neq__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
    <=> ? [Y5: A,Ys2: list(A)] : Xsa = aa(list(A),list(A),cons(A,Y5),Ys2) ) ).

% neq_Nil_conv
tff(fact_6189_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xsa: list(A),Ysa: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [X4: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),cons(A,X4),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),cons(B,Y3),Ys3))
         => ( ! [X4: 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),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xsa),Ysa) ) ) ) ) ).

% list_induct2'
tff(fact_6190_list__nonempty__induct,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(list(A),$o)] :
      ( ( Xsa != nil(A) )
     => ( ! [X4: A] : aa(list(A),$o,P,aa(list(A),list(A),cons(A,X4),nil(A)))
       => ( ! [X4: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),cons(A,X4),Xs2)) ) )
         => aa(list(A),$o,P,Xsa) ) ) ) ).

% list_nonempty_induct
tff(fact_6191_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : subseqs(A,nil(A)) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_6192_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xsa: list(A)] : n_lists(A,zero_zero(nat),Xsa) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6193_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ysa: list(B),P: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ysa) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
       => ( ! [X4: 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),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xsa),Ysa) ) ) ) ).

% list_induct2
tff(fact_6194_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xsa: list(A),Ysa: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ysa) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ysa) = 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),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X4: A,Xs2: list(A),Y3: B,Ys3: list(B),Z: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$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),$o)),P,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)),aa(list(C),list(C),cons(C,Z),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xsa),Ysa),Zs) ) ) ) ) ).

% list_induct3
tff(fact_6195_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Xsa: list(A),Ysa: list(B),Zs: list(C),Ws: list(D6),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ysa) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ysa) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D6),nat,size_size(list(D6)),Ws) )
         => ( aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,nil(A)),nil(B)),nil(C)),nil(D6))
           => ( ! [X4: A,Xs2: list(A),Y3: B,Ys3: list(B),Z: C,Zs2: list(C),W: D6,Ws2: list(D6)] :
                  ( ( 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(D6),nat,size_size(list(D6)),Ws2) )
                     => ( aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,Xs2),Ys3),Zs2),Ws2)
                       => aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)),aa(list(C),list(C),cons(C,Z),Zs2)),aa(list(D6),list(D6),cons(D6,W),Ws2)) ) ) ) )
             => aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,Xsa),Ysa),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_6196_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_6197_distinct__singleton,axiom,
    ! [A: $tType,X: A] : distinct(A,aa(list(A),list(A),cons(A,X),nil(A))) ).

% distinct_singleton
tff(fact_6198_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),X: A] : sorted_wrt(A,P,aa(list(A),list(A),cons(A,X),nil(A))) ).

% sorted_wrt1
tff(fact_6199_list__update__code_I1_J,axiom,
    ! [A: $tType,I: nat,Y: A] : list_update(A,nil(A),I,Y) = nil(A) ).

% list_update_code(1)
tff(fact_6200_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I: nat,V2: A] : list_update(A,nil(A),I,V2) = nil(A) ).

% list_update.simps(1)
tff(fact_6201_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : remove1(A,X,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_6202_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu2: list(B)] : product(A,B,nil(A),Uu2) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_6203_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_6204_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( Xsa = nil(A) )
     => ( ( Ysa = nil(A) )
       => aa(set(list(A)),$o,member(list(A),nil(A)),shuffles(A,Xsa,Ysa)) ) ) ).

% Nil_in_shufflesI
tff(fact_6205_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_6206_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_6207_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xsa: list(A)] : shuffles(A,Xsa,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xsa),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_6208_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ysa: list(A)] : shuffles(A,nil(A),Ysa) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ysa),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_6209_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_6210_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_6211_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_6212_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6213_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_6214_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ysa: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
     => ( ( ( Zs = Xsa )
         => ( Ysa != nil(A) ) )
       => ( ( ( Zs = Ysa )
           => ( Xsa != nil(A) ) )
         => ( ! [X4: A,Xs3: list(A)] :
                ( ( Xsa = aa(list(A),list(A),cons(A,X4),Xs3) )
               => ! [Z: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),cons(A,Z),Zs4) )
                   => ( ( X4 = Z )
                     => ~ aa(set(list(A)),$o,member(list(A),Zs4),shuffles(A,Xs3,Ysa)) ) ) )
           => ~ ! [Y3: A,Ys4: list(A)] :
                  ( ( Ysa = aa(list(A),list(A),cons(A,Y3),Ys4) )
                 => ! [Z: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),cons(A,Z),Zs4) )
                     => ( ( Y3 = Z )
                       => ~ aa(set(list(A)),$o,member(list(A),Zs4),shuffles(A,Xsa,Ys4)) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_6215_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(list(A),nat,size_list(A,X),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_6216_Bex__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
            & aa(A,$o,P,X2) )
      <=> finite_fold(A,$o,aTP_Lamp_sl(fun(A,$o),fun(A,fun($o,$o)),P),$false,A3) ) ) ).

% Bex_fold
tff(fact_6217_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu2: fun(A,$o)] : find(A,Uu2,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_6218_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_6219_num__of__nat_Osimps_I1_J,axiom,
    aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6220_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,X),nil(A))) ) ).

% sorted1
tff(fact_6221_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_sm(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_6222_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_6223_the__elem__set,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),nil(A)))) = X ).

% the_elem_set
tff(fact_6224_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_6225_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),one_one(nat))
     => ( aa(nat,num,num_of_nat,Na) = one2 ) ) ).

% num_of_nat_One
tff(fact_6226_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xsa != nil(A) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xsa))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xsa))) ) ) ) ).

% sum_list_strict_mono
tff(fact_6227_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6228_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A12: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A12))
     => ( ! [Ys3: list(A)] :
            ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys3) )
       => ( ! [Xs2: list(A)] :
              ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),nil(A)) )
         => ( ! [X4: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),aa(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),cons(A,Y3),Ys3))
                 => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X4),Xs2)),Ys3)
                   => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,Y3),Ys3)) ) ) )
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A12) ) ) ) ) ).

% shuffles.pinduct
tff(fact_6229_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),nil(A)))
     => ( shuffles(A,Xsa,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xsa),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_6230_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ysa: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ysa))
     => ( shuffles(A,nil(A),Ysa) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ysa),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_6231_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,Na)) = $ite(Na = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6232_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Na)) = bit0(aa(nat,num,num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_6233_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,M)),aa(nat,num,num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6234_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),cons(A,X4),Xs2) )
               => ! [Y3: A,Ys3: list(A)] :
                    ( ( Xa = aa(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)),image(list(A),list(A),cons(A,X4)),shuffles(A,Xs2,aa(list(A),list(A),cons(A,Y3),Ys3)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),cons(A,Y3)),shuffles(A,aa(list(A),list(A),cons(A,X4),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_6235_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_6236_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_6237_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_sn(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_6238_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa(nat,num,num_of_nat,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na),inc(aa(nat,num,num_of_nat,Na)),one2) ).

% num_of_nat.simps(2)
tff(fact_6239_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))),insert(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_so(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_6240_transpose__rectangle,axiom,
    ! [A: $tType,Xsa: list(list(A)),Na: nat] :
      ( ( ( Xsa = nil(list(A)) )
       => ( Na = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),I2)) = Na ) )
       => ( transpose(A,Xsa) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_sq(list(list(A)),fun(nat,list(A)),Xsa)),upt(zero_zero(nat),Na)) ) ) ) ).

% transpose_rectangle
tff(fact_6241_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss2: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) = transpose(A,Xss2) ).

% transpose.simps(2)
tff(fact_6242_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_6243_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xsa)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),transpose(B,Xsa)) ).

% transpose_map_map
tff(fact_6244_transpose__empty,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( ( transpose(A,Xsa) = nil(list(A)) )
    <=> ! [X2: list(A)] :
          ( aa(set(list(A)),$o,member(list(A),X2),aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
         => ( X2 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_6245_length__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_sr(list(A),fun(nat,nat)),Xsa),zero_zero(nat)) ).

% length_transpose
tff(fact_6246_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_6247_length__transpose__sorted,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xsa)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)) = $ite(Xsa = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_6248_rev__rev__ident,axiom,
    ! [A: $tType,Xsa: list(A)] : rev(A,rev(A,Xsa)) = Xsa ).

% rev_rev_ident
tff(fact_6249_rev__is__rev__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( rev(A,Xsa) = rev(A,Ysa) )
    <=> ( Xsa = Ysa ) ) ).

% rev_is_rev_conv
tff(fact_6250_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( nil(A) = rev(A,Xsa) )
    <=> ( Xsa = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_6251_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( rev(A,Xsa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_6252_set__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(A),set(A),set2(A),rev(A,Xsa)) = aa(list(A),set(A),set2(A),Xsa) ).

% set_rev
tff(fact_6253_length__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(A),nat,size_size(list(A)),rev(A,Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ).

% length_rev
tff(fact_6254_distinct__rev,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( distinct(A,rev(A,Xsa))
    <=> distinct(A,Xsa) ) ).

% distinct_rev
tff(fact_6255_rev__replicate,axiom,
    ! [A: $tType,Na: nat,X: A] : rev(A,replicate(A,Na,X)) = replicate(A,Na,X) ).

% rev_replicate
tff(fact_6256_singleton__rev__conv,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),cons(A,X),nil(A)) = rev(A,Xsa) )
    <=> ( aa(list(A),list(A),cons(A,X),nil(A)) = Xsa ) ) ).

% singleton_rev_conv
tff(fact_6257_rev__singleton__conv,axiom,
    ! [A: $tType,Xsa: list(A),X: A] :
      ( ( rev(A,Xsa) = aa(list(A),list(A),cons(A,X),nil(A)) )
    <=> ( Xsa = aa(list(A),list(A),cons(A,X),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_6258_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : rev(A,nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_6259_rev__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : rev(A,aa(list(B),list(A),map(B,A,F2),Xsa)) = aa(list(B),list(A),map(B,A,F2),rev(B,Xsa)) ).

% rev_map
tff(fact_6260_rev__swap,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( rev(A,Xsa) = Ysa )
    <=> ( Xsa = rev(A,Ysa) ) ) ).

% rev_swap
tff(fact_6261_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( sorted_wrt(A,P,rev(A,Xsa))
    <=> sorted_wrt(A,aTP_Lamp_ss(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xsa) ) ).

% sorted_wrt_rev
tff(fact_6262_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xsa: list(B),A2: A] : aa(A,A,foldr(B,A,F2,Xsa),A2) = foldl(A,B,aTP_Lamp_st(fun(B,fun(A,A)),fun(A,fun(B,A)),F2),A2,rev(B,Xsa)) ).

% foldr_conv_foldl
tff(fact_6263_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),A2: A,Xsa: list(B)] : foldl(A,B,F2,A2,Xsa) = aa(A,A,foldr(B,A,aTP_Lamp_su(fun(A,fun(B,A)),fun(B,fun(A,A)),F2),rev(B,Xsa)),A2) ).

% foldl_conv_foldr
tff(fact_6264_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A3: set(A),As2: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),cons(set(A),A3),As2)) = set_Cons(A,A3,listset(A,As2)) ).

% listset.simps(2)
tff(fact_6265_sorted__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xsa)))) ).

% sorted_transpose
tff(fact_6266_rev__nth,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,rev(A,Xsa)),Na) = aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xsa)),aa(nat,nat,suc,Na))) ) ) ).

% rev_nth
tff(fact_6267_rev__update,axiom,
    ! [A: $tType,K: nat,Xsa: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( rev(A,list_update(A,Xsa,K,Y)) = list_update(A,rev(A,Xsa),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xsa)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_6268_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xsa))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xsa),I4)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_6269_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xsa))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),J)),aa(nat,A,nth(A,Xsa),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_6270_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xsa))
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),J3)),aa(nat,A,nth(A,Xsa),I4)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_6271_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xsa))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xsa),Y) = $ite(Xsa = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xsa),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_6272_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xsa: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_sv(nat,fun(list(A),$o),I)),Xsa)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xsa)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xsa),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_6273_transpose__column,axiom,
    ! [A: $tType,Xsa: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_sw(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_sv(nat,fun(list(A),$o),I)),transpose(A,Xsa))) = aa(nat,list(A),nth(list(A),Xsa),I) ) ) ) ).

% transpose_column
tff(fact_6274_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xsa: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xsa)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xsa) ).

% filter_filter
tff(fact_6275_filter__True,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X4) )
     => ( aa(list(A),list(A),filter2(A,P),Xsa) = Xsa ) ) ).

% filter_True
tff(fact_6276_remove1__filter__not,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xsa: list(A)] :
      ( ~ aa(A,$o,P,X)
     => ( remove1(A,X,aa(list(A),list(A),filter2(A,P),Xsa)) = aa(list(A),list(A),filter2(A,P),Xsa) ) ) ).

% remove1_filter_not
tff(fact_6277_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xsa)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_sy(fun(A,$o),fun(list(A),fun(A,$o)),P),Xsa)) ).

% set_filter
tff(fact_6278_filter__False,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,P,X4) )
     => ( aa(list(A),list(A),filter2(A,P),Xsa) = nil(A) ) ) ).

% filter_False
tff(fact_6279_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_6280_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xsa) = nil(A) )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,P,X2) ) ) ).

% filter_empty_conv
tff(fact_6281_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P),Xsa) )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,P,X2) ) ) ).

% empty_filter_conv
tff(fact_6282_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Na: nat,X: A] :
      aa(list(A),list(A),filter2(A,P),replicate(A,Na,X)) = $ite(aa(A,$o,P,X),replicate(A,Na,X),nil(A)) ).

% filter_replicate
tff(fact_6283_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),P: fun(B,$o)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xsa))
     => distinct(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xsa))) ) ).

% distinct_map_filter
tff(fact_6284_rev__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] : rev(A,aa(list(A),list(A),filter2(A,P),Xsa)) = aa(list(A),list(A),filter2(A,P),rev(A,Xsa)) ).

% rev_filter
tff(fact_6285_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),X)),Xsa)),X) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),X)),Xsa) ).

% replicate_length_filter
tff(fact_6286_partition__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] : aa(set(list(A)),$o,member(list(A),Xsa),shuffles(A,aa(list(A),list(A),filter2(A,P),Xsa),aa(list(A),list(A),filter2(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)),Xsa))) ).

% partition_in_shuffles
tff(fact_6287_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: 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),Xsa))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)),Xsa))) = aa(list(A),nat,size_size(list(A)),Xsa) ).

% sum_length_filter_compl
tff(fact_6288_inter__set__filter,axiom,
    ! [A: $tType,A3: set(A),Xsa: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3)),Xsa)) ).

% inter_set_filter
tff(fact_6289_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xsa) = Xsa )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X2) ) ) ).

% filter_id_conv
tff(fact_6290_filter__cong,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xsa = Ysa )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Ysa))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( aa(list(A),list(A),filter2(A,P),Xsa) = aa(list(A),list(A),filter2(A,Q),Ysa) ) ) ) ).

% filter_cong
tff(fact_6291_distinct__filter,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( distinct(A,Xsa)
     => distinct(A,aa(list(A),list(A),filter2(A,P),Xsa)) ) ).

% distinct_filter
tff(fact_6292_sorted__wrt__filter,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xsa: list(A),P: fun(A,$o)] :
      ( sorted_wrt(A,F2,Xsa)
     => sorted_wrt(A,F2,aa(list(A),list(A),filter2(A,P),Xsa)) ) ).

% sorted_wrt_filter
tff(fact_6293_filter__remove1,axiom,
    ! [A: $tType,Q: fun(A,$o),X: A,Xsa: list(A)] : aa(list(A),list(A),filter2(A,Q),remove1(A,X,Xsa)) = remove1(A,X,aa(list(A),list(A),filter2(A,Q),Xsa)) ).

% filter_remove1
tff(fact_6294_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: 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),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_filter_le
tff(fact_6295_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: 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),Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% filter_is_subset
tff(fact_6296_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xsa: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_sz(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xsa)),Xsa)) ) ).

% sorted_same
tff(fact_6297_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xsa: list(A)] :
      aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),cons(A,X),Xsa)) = $ite(aa(A,$o,P,X),aa(list(A),list(A),cons(A,X),aa(list(A),list(A),filter2(A,P),Xsa)),aa(list(A),list(A),filter2(A,P),Xsa)) ).

% filter.simps(2)
tff(fact_6298_filter__shuffles,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A),Ysa: list(A)] : aa(set(list(A)),set(list(A)),image(list(A),list(A),filter2(A,P)),shuffles(A,Xsa,Ysa)) = shuffles(A,aa(list(A),list(A),filter2(A,P),Xsa),aa(list(A),list(A),filter2(A,P),Ysa)) ).

% filter_shuffles
tff(fact_6299_length__filter__less,axiom,
    ! [A: $tType,X: A,Xsa: list(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
     => ( ~ aa(A,$o,P,X)
       => 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),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% length_filter_less
tff(fact_6300_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xsa))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xsa))) ) ) ).

% sorted_filter
tff(fact_6301_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xsa: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_ta(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F2),G),Xsa)),Xsa))) ) ).

% sorted_map_same
tff(fact_6302_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,$o),Xsa: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xsa))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_tb(fun(B,A),fun(fun(B,$o),fun(B,A)),F2),P)),Xsa)) ) ).

% sum_list_map_filter'
tff(fact_6303_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,$o),Xsa: 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),Xsa)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xsa))) ).

% sum_list_filter_le_nat
tff(fact_6304_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xsa: list(A),P: fun(A,$o),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xsa))
             => ( ~ aa(A,$o,P,X4)
               => ( aa(A,B,F2,X4) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),filter2(A,P),Xsa))) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xsa)) ) ) ) ).

% sum_list_map_filter
tff(fact_6305_set__minus__filter__out,axiom,
    ! [A: $tType,Xsa: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_tc(A,fun(A,$o),Y)),Xsa)) ).

% set_minus_filter_out
tff(fact_6306_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_td(list(A),fun(A,$o),Xsa)),Zs) = Ysa ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_6307_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_te(list(A),fun(A,$o),Xsa)),Zs) = Xsa ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_6308_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_td(list(A),fun(A,$o),Ysa)),Zs) = Xsa ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_6309_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ysa)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xsa,Ysa))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_te(list(A),fun(A,$o),Ysa)),Zs) = Ysa ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_6310_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xsa: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P3),Xsa)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_tf(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xsa))) ).

% length_filter_conv_card
tff(fact_6311_distinct__length__filter,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( distinct(A,Xsa)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xsa)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(list(A),set(A),set2(A),Xsa))) ) ) ).

% distinct_length_filter
tff(fact_6312_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Xss2: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xsa))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_tg(list(B),fun(nat,nat)),Xss2),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_th(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_ti(list(B),$o)),Xss2)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_6313_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xsa: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xsa)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_sw(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_sv(nat,fun(list(A),$o),I)),Xsa)) ) ) ).

% nth_transpose
tff(fact_6314_transpose__max__length,axiom,
    ! [A: $tType,Xsa: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_sr(list(A),fun(nat,nat)),transpose(A,Xsa)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_tj(list(A),$o)),Xsa)) ).

% transpose_max_length
tff(fact_6315_transpose__column__length,axiom,
    ! [A: $tType,Xsa: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_sv(nat,fun(list(A),$o),I)),transpose(A,Xsa))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),I)) ) ) ) ).

% transpose_column_length
tff(fact_6316_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o),Xsa: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xsa)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_tk(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F2),P),Xsa) ).

% map_filter_map_filter
tff(fact_6317_transpose__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xsa)))
     => ( transpose(A,transpose(A,Xsa)) = takeWhile(list(A),aTP_Lamp_tj(list(A),$o),Xsa) ) ) ).

% transpose_transpose
tff(fact_6318_takeWhile__idem,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] : takeWhile(A,P,takeWhile(A,P,Xsa)) = takeWhile(A,P,Xsa) ).

% takeWhile_idem
tff(fact_6319_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( ( takeWhile(A,P,Xsa) = Xsa )
    <=> ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X2) ) ) ).

% takeWhile_eq_all_conv
tff(fact_6320_takeWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Na: nat,X: A] :
      takeWhile(A,P,replicate(A,Na,X)) = $ite(aa(A,$o,P,X),replicate(A,Na,X),nil(A)) ).

% takeWhile_replicate
tff(fact_6321_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xsa: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),cons(A,X),Xsa)) = $ite(aa(A,$o,P,X),aa(list(A),list(A),cons(A,X),takeWhile(A,P,Xsa)),nil(A)) ).

% takeWhile.simps(2)
tff(fact_6322_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xsa)) ) ) ).

% sorted_takeWhile
tff(fact_6323_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: 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,Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_takeWhile_le
tff(fact_6324_distinct__takeWhile,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( distinct(A,Xsa)
     => distinct(A,takeWhile(A,P,Xsa)) ) ).

% distinct_takeWhile
tff(fact_6325_set__takeWhileD,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),Xsa: list(A)] :
      ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),takeWhile(A,P,Xsa)))
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xsa))
        & aa(A,$o,P,X) ) ) ).

% set_takeWhileD
tff(fact_6326_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K )
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( takeWhile(A,P,L) = takeWhile(A,Q,K) ) ) ) ).

% takeWhile_cong
tff(fact_6327_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : takeWhile(A,P,nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_6328_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_6329_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xsa: 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,Xsa)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xsa)),J) = aa(nat,A,nth(A,Xsa),J) ) ) ).

% takeWhile_nth
tff(fact_6330_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xsa))),aa(list(A),nat,size_size(list(A)),Xsa))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xsa),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xsa)))) ) ).

% nth_length_takeWhile
tff(fact_6331_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,Xsa: list(B)] : map_filter(B,A,F2,aa(list(B),list(B),cons(B,X),Xsa)) = case_option(list(A),A,map_filter(B,A,F2,Xsa),aa(list(B),fun(A,list(A)),aTP_Lamp_tl(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xsa),aa(B,option(A),F2,X)) ).

% map_filter_simps(1)
tff(fact_6332_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xsa: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xsa),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xsa))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_6333_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F2),Xsa)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,$o)),F2),T2)),Xsa) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,$o)),F2),T2),Xsa) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_6334_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_tn(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_6335_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6336_member__filter,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),A3: set(A)] :
      ( aa(set(A),$o,member(A,X),filter3(A,P,A3))
    <=> ( aa(set(A),$o,member(A,X),A3)
        & aa(A,$o,P,X) ) ) ).

% member_filter
tff(fact_6337_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter3(A,P,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_to(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_6338_filter__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xsa: list(A)] : filter3(A,P,aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xsa)) ).

% filter_set
tff(fact_6339_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,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3),B3) ) ) ).

% inter_Set_filter
tff(fact_6340_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6341_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa),aa(list(int),list(int),cons(int,X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)),nil(int)) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_6342_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_6343_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_6344_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6345_upto__single,axiom,
    ! [I: int] : upto(I,I) = aa(list(int),list(int),cons(int,I),nil(int)) ).

% upto_single
tff(fact_6346_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6347_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,J),I)),one_one(int))) ).

% length_upto
tff(fact_6348_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6349_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6350_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6351_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6352_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).

% upto_code
tff(fact_6353_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt(int,ord_less_eq(int),upto(M,Na)) ).

% sorted_upto
tff(fact_6354_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_6355_distinct__upto,axiom,
    ! [I: int,J: int] : distinct(int,upto(I,J)) ).

% distinct_upto
tff(fact_6356_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_6357_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6358_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6359_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa),aa(list(int),list(int),cons(int,X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)),nil(int)) ) ) ).

% upto.elims
tff(fact_6360_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6361_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A12: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A12))
     => ( ! [Ys3: list(A)] :
            ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys3) )
       => ( ! [X4: A,Xs2: list(A),Ys3: list(A)] :
              ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),Ys3))
             => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Ys3),Xs2)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X4),Xs2)),Ys3) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A12) ) ) ) ).

% splice.pinduct
tff(fact_6362_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_tp(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Sup_fin.eq_fold'
tff(fact_6363_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_6364_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,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_6365_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6366_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6367_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_6368_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_6369_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6370_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$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,lattic5882676163264333800up_fin(A),A3)),X) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6371_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),X)
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A11),X) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6372_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_6373_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_6374_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X4: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),Y3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X4)),aa(A,A,H,Y3))
         => ( aa(set(A),$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),image(A,A,H),N3)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_6375_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_6376_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)) )
           => ( ! [X4: A,Y3: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),A3) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6377_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,X),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6378_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_6379_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,sup_sup(A),X,A3) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6380_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6381_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6382_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa) = Y )
     => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa )
             => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ~ ! [X4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),cons(A,X4),Xs2) )
               => ( ( Y = aa(list(A),list(A),cons(A,X4),splice(A,Xa,Xs2)) )
                 => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X4),Xs2)),Xa)) ) ) ) ) ) ).

% splice.pelims
tff(fact_6383_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_tq(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Inf_fin.eq_fold'
tff(fact_6384_split__Nil__iff,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] :
      ( ( splice(A,Xsa,Ysa) = nil(A) )
    <=> ( ( Xsa = nil(A) )
        & ( Ysa = nil(A) ) ) ) ).

% split_Nil_iff
tff(fact_6385_splice__Nil2,axiom,
    ! [A: $tType,Xsa: list(A)] : splice(A,Xsa,nil(A)) = Xsa ).

% splice_Nil2
tff(fact_6386_splice__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] : aa(set(list(A)),$o,member(list(A),splice(A,Xsa,Ysa)),shuffles(A,Xsa,Ysa)) ).

% splice_in_shuffles
tff(fact_6387_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_6388_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,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_6389_length__splice,axiom,
    ! [A: $tType,Xsa: list(A),Ysa: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xsa,Ysa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ysa)) ).

% length_splice
tff(fact_6390_splice__replicate,axiom,
    ! [A: $tType,M: nat,X: A,Na: nat] : splice(A,replicate(A,M,X),replicate(A,Na,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),X) ).

% splice_replicate
tff(fact_6391_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$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)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6392_splice_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Ysa: list(A)] : splice(A,aa(list(A),list(A),cons(A,X),Xsa),Ysa) = aa(list(A),list(A),cons(A,X),splice(A,Ysa,Xsa)) ).

% splice.simps(2)
tff(fact_6393_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_6394_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_6395_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6396_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ysa: list(A)] : splice(A,nil(A),Ysa) = Ysa ).

% splice.simps(1)
tff(fact_6397_splice_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != Xa ) )
       => ~ ! [X4: A,Xs2: list(A)] :
              ( ( X = aa(list(A),list(A),cons(A,X4),Xs2) )
             => ( Y != aa(list(A),list(A),cons(A,X4),splice(A,Xa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_6398_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A11) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6399_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6400_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6401_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_6402_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_6403_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X4: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),Y3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X4)),aa(A,A,H,Y3))
         => ( aa(set(A),$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),image(A,A,H),N3)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_6404_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_6405_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,X),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6406_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)) )
           => ( ! [X4: A,Y3: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A3) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6407_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_6408_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_6409_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,inf_inf(A),X,A3) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6410_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6411_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6412_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Ysa: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xsa)),Ysa))
     => ( splice(A,aa(list(A),list(A),cons(A,X),Xsa),Ysa) = aa(list(A),list(A),cons(A,X),splice(A,Ysa,Xsa)) ) ) ).

% splice.psimps(2)
tff(fact_6413_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ysa: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ysa))
     => ( splice(A,nil(A),Ysa) = Ysa ) ) ).

% splice.psimps(1)
tff(fact_6414_divmod__step__integer__def,axiom,
    ! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_tr(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_6415_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)) )
           => ( linord4507533701916653071of_set(A,A3) = aa(list(A),list(A),cons(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3)),linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6416_sorted__list__of__set__range,axiom,
    ! [M: nat,Na: nat] : linord4507533701916653071of_set(nat,set_or7035219750837199246ssThan(nat,M,Na)) = upt(M,Na) ).

% sorted_list_of_set_range
tff(fact_6417_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_6418_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( linord4507533701916653071of_set(A,A3) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_6419_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),linord4507533701916653071of_set(A,A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_6420_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)),linord4507533701916653071of_set(A,A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_6421_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)
         => ( ( 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_6422_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,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_6423_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( linord4507533701916653071of_set(A,A3) = linord4507533701916653071of_set(A,B3) )
         => ( aa(set(A),$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_6424_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_6425_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(code_integer,M,Na) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Na))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),Na))) ).

% divmod_integer'_def
tff(fact_6426_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : distinct(A,linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_6427_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_6428_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_6429_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_6430_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_6431_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( distinct(A,Xsa)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xsa)) = Xsa ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_6432_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_6433_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) ) )
          <=> ( linord4507533701916653071of_set(A,A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_6434_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(aa(int,int,aa(int,fun(int,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_6435_Code__Numeral_Opositive__def,axiom,
    code_positive = numeral_numeral(code_integer) ).

% Code_Numeral.positive_def
tff(fact_6436_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_6437_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_6438_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_6439_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),X) ) ).

% less_integer.abs_eq
tff(fact_6440_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),X) ) ).

% less_eq_integer.abs_eq
tff(fact_6441_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      aa(num,code_integer,code_integer_of_num,aa(num,num,bit1,Na)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_6442_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_ts(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_6443_int__of__integer__of__nat,axiom,
    ! [Na: nat] : code_int_of_integer(aa(nat,code_integer,semiring_1_of_nat(code_integer),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ).

% int_of_integer_of_nat
tff(fact_6444_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_6445_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_6446_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(X)),code_int_of_integer(Xa)) ) ).

% less_integer.rep_eq
tff(fact_6447_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_6448_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(X)),code_int_of_integer(Xa)) ) ).

% less_eq_integer.rep_eq
tff(fact_6449_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_6450_integer__of__num__def,axiom,
    code_integer_of_num = numeral_numeral(code_integer) ).

% integer_of_num_def
tff(fact_6451_integer__of__num__triv_I1_J,axiom,
    aa(num,code_integer,code_integer_of_num,one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_6452_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      aa(num,code_integer,code_integer_of_num,bit0(Na)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_6453_integer__of__num__triv_I2_J,axiom,
    aa(num,code_integer,code_integer_of_num,bit0(one2)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% integer_of_num_triv(2)
tff(fact_6454_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,num,code_num_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_tt(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% num_of_integer_code
tff(fact_6455_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_tu(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_6456_nat__of__integer__of__nat,axiom,
    ! [Na: nat] : code_nat_of_integer(aa(nat,code_integer,semiring_1_of_nat(code_integer),Na)) = Na ).

% nat_of_integer_of_nat
tff(fact_6457_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),code_nat_of_integer(K)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).

% of_nat_of_integer
tff(fact_6458_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_6459_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_6460_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_6461_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),~ dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)),K)) ).

% bit_cut_integer_def
tff(fact_6462_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_6463_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ig(nat,nat),Nat) ).

% pred_def
tff(fact_6464_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_6465_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,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_tv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_6466_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_tw(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_6467_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_6468_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),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_tx(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),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ty(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_6469_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S2))
           => ( infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_tz(set(A),fun(nat,fun(A,$o)),S2),Na)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_6470_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,M)),infini527867602293511546merate(A,S2,Na))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% enumerate_mono_iff
tff(fact_6471_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(set(A),nat,finite_card(A),S2))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,M)),infini527867602293511546merate(A,S2,Na))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_6472_le__enumerate,axiom,
    ! [S2: set(nat),Na: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),infini527867602293511546merate(nat,S2,Na)) ) ).

% le_enumerate
tff(fact_6473_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] : infini527867602293511546merate(A,S2,zero_zero(nat)) = ord_Least(A,aTP_Lamp_ua(set(A),fun(A,$o),S2)) ) ).

% enumerate_0
tff(fact_6474_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,Na)),infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na))) ) ) ).

% enumerate_step
tff(fact_6475_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( ~ aa(set(A),$o,finite_finite2(A),S2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,M)),infini527867602293511546merate(A,S2,Na)) ) ) ) ).

% enumerate_mono
tff(fact_6476_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
           => aa(set(A),$o,member(A,infini527867602293511546merate(A,S2,Na)),S2) ) ) ) ).

% finite_enumerate_in_set
tff(fact_6477_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),S: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),$o,member(A,S),S2)
           => ? [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2))
                & ( infini527867602293511546merate(A,S2,N) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_6478_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y4: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X5))
             => ( infini527867602293511546merate(A,X5,I2) = infini527867602293511546merate(A,Y4,I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X5)
           => ( aa(set(A),$o,finite_finite2(A),Y4)
             => ( ( aa(set(A),nat,finite_card(A),X5) = aa(set(A),nat,finite_card(A),Y4) )
               => ( X5 = Y4 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_6479_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,M)),infini527867602293511546merate(A,S2,Na)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_6480_finite__le__enumerate,axiom,
    ! [S2: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(nat),nat,finite_card(nat),S2))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),infini527867602293511546merate(nat,S2,Na)) ) ) ).

% finite_le_enumerate
tff(fact_6481_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_tz(set(A),fun(nat,fun(A,$o)),S2),Na)) ) ) ) ).

% enumerate_Suc''
tff(fact_6482_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S2,Na)),infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na))) ) ) ) ).

% finite_enumerate_step
tff(fact_6483_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] : infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na)) = infini527867602293511546merate(A,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),infini527867602293511546merate(A,S2,zero_zero(nat))),bot_bot(set(A)))),Na) ) ).

% enumerate_Suc'
tff(fact_6484_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y4: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X5))
             => ( infini527867602293511546merate(A,X5,I2) = infini527867602293511546merate(A,Y4,I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X5)
           => ( aa(set(A),$o,finite_finite2(A),Y4)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X5)),aa(set(A),nat,finite_card(A),Y4))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),Y4) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_6485_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat,S: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(A,set(A),set_ord_lessThan(A),S))))
           => ( 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)),Na) = infini527867602293511546merate(A,S2,Na) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_6486_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] : infini527867602293511546merate(A,S2,aa(nat,nat,suc,Na)) = infini527867602293511546merate(A,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),ord_Least(A,aTP_Lamp_ua(set(A),fun(A,$o),S2))),bot_bot(set(A)))),Na) ) ).

% enumerate_Suc
tff(fact_6487_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_tw(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_6488_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V2: num,Na: 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)),Na)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Na)),aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Na))) ) ).

% rec_nat_add_eq_if
tff(fact_6489_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_6490_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_6491_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_6492_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_6493_old_Orec__nat__def,axiom,
    ! [A: $tType,X3: A,Xa3: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X3,Xa3),Xb2) = the(A,rec_set_nat(A,X3,Xa3,Xb2)) ).

% old.rec_nat_def
tff(fact_6494_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_6495_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),$o),ord_less_eq(set(A)),B3),A3)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),B3)
           => ( aa(A,$o,Q,X4)
             => aa(A,$o,P,X4) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),B3),Q))),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% subset_CollectI
tff(fact_6496_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),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),A3),P)))
      <=> ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),B3)
           => aa(A,$o,P,X2) ) ) ) ).

% subset_Collect_iff
tff(fact_6497_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),Na: nat] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,aa(nat,nat,suc,Na)) = aa(A,A,aa(nat,fun(A,A),F22,Na),aa(nat,A,F2,Na)) ) ) ).

% rec_nat_Suc_imp
tff(fact_6498_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_6499_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_6500_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se4197421643247451524op_bit(A,Na,zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_6501_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : bit_se4197421643247451524op_bit(A,M,bit_se4197421643247451524op_bit(A,Na,A2)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),A2) ) ).

% drop_bit_drop_bit
tff(fact_6502_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,Na,A2)),bit_se4197421643247451524op_bit(A,Na,B2)) ) ).

% drop_bit_and
tff(fact_6503_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4197421643247451524op_bit(A,Na,A2)),bit_se4197421643247451524op_bit(A,Na,B2)) ) ).

% drop_bit_or
tff(fact_6504_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Na,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4197421643247451524op_bit(A,Na,A2)),bit_se4197421643247451524op_bit(A,Na,B2)) ) ).

% drop_bit_xor
tff(fact_6505_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,B2: $o] :
          bit_se4197421643247451524op_bit(A,Na,aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_6506_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4197421643247451524op_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_6507_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4197421643247451524op_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_6508_drop__bit__minus__one,axiom,
    ! [Na: nat] : bit_se4197421643247451524op_bit(int,Na,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_6509_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Na),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,Na,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_6510_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Na),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,Na,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_6511_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se4197421643247451524op_bit(A,Na,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_6512_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_6513_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_6514_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = bit_se4197421643247451524op_bit(int,Na,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_6515_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_6516_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,Na,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_6517_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : bit_se4197421643247451524op_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Na),M)),bit_se4197421643247451524op_bit(A,M,A2)) ) ).

% drop_bit_take_bit
tff(fact_6518_drop__bit__push__bit__int,axiom,
    ! [M: nat,Na: nat,K: int] : bit_se4197421643247451524op_bit(int,M,bit_se4730199178511100633sh_bit(int,Na,K)) = bit_se4197421643247451524op_bit(int,aa(nat,nat,minus_minus(nat,M),Na),bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,Na),M),K)) ).

% drop_bit_push_bit_int
tff(fact_6519_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,Na,A2)) = bit_se4197421643247451524op_bit(A,Na,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),A2)) ) ).

% take_bit_drop_bit
tff(fact_6520_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2) = A2 )
        <=> ( bit_se4197421643247451524op_bit(A,Na,A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_6521_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,M,Na)) = bit_se4197421643247451524op_bit(A,M,aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_drop_bit
tff(fact_6522_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,M: nat] : bit_se4197421643247451524op_bit(A,Na,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,Na,M)) ) ).

% drop_bit_of_nat
tff(fact_6523_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),bit_se4730199178511100633sh_bit(A,Na,one_one(A))) = bit_se4197421643247451524op_bit(A,Na,A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_6524_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,Na,A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_6525_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,Na,bit_se4197421643247451524op_bit(A,Na,A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A2)) = A2 ) ).

% bits_ident
tff(fact_6526_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4197421643247451524op_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),bit_se4197421643247451524op_bit(A,Na,A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_6527_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( bit_se4197421643247451524op_bit(A,Na,A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_6528_drop__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se4197421643247451524op_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% drop_bit_int_def
tff(fact_6529_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Na),A2) = bit_se4197421643247451524op_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_6530_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se4197421643247451524op_bit(A,Na,A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% drop_bit_eq_div
tff(fact_6531_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se4197421643247451524op_bit(A,Na,A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_6532_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se4197421643247451524op_bit(A,Na,A2))
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Na) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_6533_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,M: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Na,aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,Na,A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)))) ) ).

% slice_eq_mask
tff(fact_6534_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          bit_se4197421643247451524op_bit(A,Na,A2) = $ite(Na = zero_zero(nat),A2,bit_se4197421643247451524op_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_6535_max__extp_Omax__extI,axiom,
    ! [A: $tType,X5: set(A),Y4: set(A),R2: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,finite_finite2(A),Y4)
       => ( ( Y4 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => ? [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),Y4)
                    & aa(A,$o,aa(A,fun(A,$o),R2,X4),Xa3) ) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),X5),Y4) ) ) ) ) ).

% max_extp.max_extI
tff(fact_6536_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : bit_se4197421643247451524op_bit(nat,Na,aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Na = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_6537_drop__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : bit_se4197421643247451524op_bit(nat,Na,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se4197421643247451524op_bit(int,Na,K)) ).

% drop_bit_nat_eq
tff(fact_6538_drop__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : bit_se4197421643247451524op_bit(nat,Na,M) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% drop_bit_nat_def
tff(fact_6539_max__extp_Ocases,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),A12),A23)
     => ~ ( aa(set(A),$o,finite_finite2(A),A12)
         => ( aa(set(A),$o,finite_finite2(A),A23)
           => ( ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
             => ~ ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A12)
                   => ? [Xa4: A] :
                        ( aa(set(A),$o,member(A,Xa4),A23)
                        & aa(A,$o,aa(A,fun(A,$o),R2,X3),Xa4) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_6540_max__extp_Osimps,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),A12),A23)
    <=> ( aa(set(A),$o,finite_finite2(A),A12)
        & aa(set(A),$o,finite_finite2(A),A23)
        & ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
        & ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A12)
           => ? [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A23)
                & aa(A,$o,aa(A,fun(A,$o),R2,X2),Xa2) ) ) ) ) ).

% max_extp.simps
tff(fact_6541_Suc__0__mod__numeral,axiom,
    ! [K: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_6542_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_6543_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_6544_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_6545_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_div_numeral
tff(fact_6546_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_mod_numeral
tff(fact_6547_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Ysa: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xsa) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ysa) )
     => ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xsa) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ysa) )
       => ( Xsa = Ysa ) ) ) ).

% pair_list_eqI
tff(fact_6548_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X5: set(A),A3: set(product_prod(A,B)),Y4: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X5 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3) )
     => ( ( Y4 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A3) )
       => ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),X5)
             => ! [Xa4: B] :
                  ( aa(set(B),$o,member(B,Xa4),Y4)
                 => ( aa(B,$o,aa(A,fun(B,$o),P,X4),Xa4)
                   => aa(B,$o,aa(A,fun(B,$o),Q,X4),Xa4) ) ) )
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_6549_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X3),Xa3) = aa(B,C,aa(A,fun(B,C),X3,aa(product_prod(A,B),A,product_fst(A,B),Xa3)),aa(product_prod(A,B),B,product_snd(A,B),Xa3)) ).

% case_prod_unfold
tff(fact_6550_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),X3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),X3) = aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),X3)),aa(product_prod(A,B),B,product_snd(A,B),X3)) ).

% case_prod_beta'
tff(fact_6551_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D6: $tType,A: $tType,F2: fun(D6,fun(B,C)),G: fun(A,D6)] : aa(fun(A,D6),fun(product_prod(A,B),C),aTP_Lamp_ub(fun(D6,fun(B,C)),fun(fun(A,D6),fun(product_prod(A,B),C)),F2),G) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D6),fun(A,fun(B,C)),aTP_Lamp_uc(fun(D6,fun(B,C)),fun(fun(A,D6),fun(A,fun(B,C))),F2),G)) ).

% split_comp_eq
tff(fact_6552_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P3: product_prod(B,A),Q: fun(C,$o),F2: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P3)),aa(product_prod(B,A),B,product_fst(B,A),P3))
     => ( ! [X4: B,Y3: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y3),X4)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F2,X4),Y3)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F2),P3)) ) ) ).

% exE_realizer
tff(fact_6553_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_ud(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_6554_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_pu(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_6555_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = the(product_prod(A,B),aTP_Lamp_ue(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_6556_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = fChoice(product_prod(A,B),aTP_Lamp_ue(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_6557_divides__aux__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Qr: product_prod(A,A)] :
          ( unique5940410009612947441es_aux(A,Qr)
        <=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).

% divides_aux_def
tff(fact_6558_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) ) ).

% fst_divmod
tff(fact_6559_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,Na)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Na)) ) ).

% snd_divmod
tff(fact_6560_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F2,G,P3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_6561_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),Na: nat,Xsa: list(A)] :
      ( aa(set(product_prod(nat,A)),$o,member(product_prod(nat,A),P3),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Na,Xsa)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(product_prod(nat,A),nat,product_fst(nat,A),P3))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Na))
        & ( aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),Na)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6562_enumerate__simps_I1_J,axiom,
    ! [A: $tType,Na: nat] : enumerate(A,Na,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_6563_length__enumerate,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Na,Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ).

% length_enumerate
tff(fact_6564_map__snd__enumerate,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,Na,Xsa)) = Xsa ).

% map_snd_enumerate
tff(fact_6565_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Na: nat,X: A,Xsa: list(A)] : enumerate(A,Na,aa(list(A),list(A),cons(A,X),Xsa)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),cons(product_prod(nat,A),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Na),X)),enumerate(A,aa(nat,nat,suc,Na),Xsa)) ).

% enumerate_simps(2)
tff(fact_6566_map__fst__enumerate,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Na,Xsa)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(list(A),nat,size_size(list(A)),Xsa))) ).

% map_fst_enumerate
tff(fact_6567_distinct__enumerate,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : distinct(product_prod(nat,A),enumerate(A,Na,Xsa)) ).

% distinct_enumerate
tff(fact_6568_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R3))) ).

% quotient_of_denom_pos'
tff(fact_6569_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).

% bezw_non_0
tff(fact_6570_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y)))))) ).

% bezw.simps
tff(fact_6571_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_6572_enumerate__map__upt,axiom,
    ! [A: $tType,Na: nat,F2: fun(nat,A),M: nat] : enumerate(A,Na,aa(list(nat),list(A),map(nat,A,F2),upt(Na,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_uf(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(Na,M)) ).

% enumerate_map_upt
tff(fact_6573_sorted__enumerate,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Na,Xsa))) ).

% sorted_enumerate
tff(fact_6574_enumerate__replicate__eq,axiom,
    ! [A: $tType,Na: nat,M: nat,A2: A] : enumerate(A,Na,replicate(A,M,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ug(A,fun(nat,product_prod(nat,A)),A2)),upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M))) ).

% enumerate_replicate_eq
tff(fact_6575_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xsa: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Na,Xsa)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),aa(nat,A,nth(A,Xsa),M)) ) ) ).

% nth_enumerate_eq
tff(fact_6576_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa)))))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_6577_one__mod__minus__numeral,axiom,
    ! [Na: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na)))) ).

% one_mod_minus_numeral
tff(fact_6578_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na))) ).

% minus_numeral_mod_numeral
tff(fact_6579_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na)))) ).

% numeral_mod_minus_numeral
tff(fact_6580_minus__one__mod__numeral,axiom,
    ! [Na: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na))) ).

% minus_one_mod_numeral
tff(fact_6581_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R3: int] :
      adjust_mod(L,R3) = $ite(R3 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,L),R3)) ).

% Divides.adjust_mod_def
tff(fact_6582_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      normalize(P3) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
        $let(
          a3: int,
          a3:= gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P3),aa(product_prod(int,int),int,product_snd(int,int),P3)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a3)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a3: int,
            a3:= aa(int,int,uminus_uminus(int),gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P3),aa(product_prod(int,int),int,product_snd(int,int),P3))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a3)) ) ) ) ).

% normalize_def
tff(fact_6583_divmod__integer__eq__cases,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),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer),times_times(code_integer)),sgn_sgn(code_integer)),L),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_uh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_6584_gcd__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : gcd_gcd(A,gcd_gcd(A,A2,B2),B2) = gcd_gcd(A,A2,B2) ) ).

% gcd_right_idem
tff(fact_6585_gcd__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : gcd_gcd(A,A2,gcd_gcd(A,A2,B2)) = gcd_gcd(A,A2,B2) ) ).

% gcd_left_idem
tff(fact_6586_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( gcd_gcd(A,A2,B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_6587_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,one_one(A),A2) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_6588_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,A2,one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_6589_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Na: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Na),Na) = gcd_gcd(A,M,Na) ) ).

% gcd_add1
tff(fact_6590_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Na: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Na)) = gcd_gcd(A,M,Na) ) ).

% gcd_add2
tff(fact_6591_gcd__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : gcd_gcd(A,A2,aa(A,A,uminus_uminus(A),B2)) = gcd_gcd(A,A2,B2) ) ).

% gcd_neg2
tff(fact_6592_gcd__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),A2),B2) = gcd_gcd(A,A2,B2) ) ).

% gcd_neg1
tff(fact_6593_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,Na: nat,B2: A] : gcd_gcd(A,aa(nat,A,power_power(A,A2),Na),aa(nat,A,power_power(A,B2),Na)) = aa(nat,A,power_power(A,gcd_gcd(A,A2,B2)),Na) ) ).

% gcd_exp
tff(fact_6594_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% map_comp_map
tff(fact_6595_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% List.map.comp
tff(fact_6596_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,comp(B,A,C,G,F2)),V2) ).

% list.map_comp
tff(fact_6597_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,comp(B,A,C,F2,G)),List) ).

% List.map.compositionality
tff(fact_6598_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Xsa: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),Xsa)) = aa(list(C),list(A),map(C,A,comp(B,A,C,F2,G)),Xsa) ).

% map_map
tff(fact_6599_gcd__dvd1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,gcd_gcd(A,A2,B2),A2) ) ).

% gcd_dvd1
tff(fact_6600_gcd__dvd2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,gcd_gcd(A,A2,B2),B2) ) ).

% gcd_dvd2
tff(fact_6601_gcd__greatest__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,gcd_gcd(A,B2,C2))
        <=> ( dvd_dvd(A,A2,B2)
            & dvd_dvd(A,A2,C2) ) ) ) ).

% gcd_greatest_iff
tff(fact_6602_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd(int,M,one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_6603_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,X,aa(int,int,uminus_uminus(int),Y)) = gcd_gcd(int,X,Y) ).

% gcd_neg2_int
tff(fact_6604_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,aa(int,int,uminus_uminus(int),X),Y) = gcd_gcd(int,X,Y) ).

% gcd_neg1_int
tff(fact_6605_abs__gcd__int,axiom,
    ! [X: int,Y: int] : aa(int,int,abs_abs(int),gcd_gcd(int,X,Y)) = gcd_gcd(int,X,Y) ).

% abs_gcd_int
tff(fact_6606_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,aa(int,int,abs_abs(int),X),Y) = gcd_gcd(int,X,Y) ).

% gcd_abs1_int
tff(fact_6607_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,X,aa(int,int,abs_abs(int),Y)) = gcd_gcd(int,X,Y) ).

% gcd_abs2_int
tff(fact_6608_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Na: num,A2: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)),A2) = gcd_gcd(A,aa(num,A,numeral_numeral(A),Na),A2) ) ).

% gcd_neg_numeral_1
tff(fact_6609_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Na: num] : gcd_gcd(A,A2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = gcd_gcd(A,A2,aa(num,A,numeral_numeral(A),Na)) ) ).

% gcd_neg_numeral_2
tff(fact_6610_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,gcd_gcd(A,A2,B2),one_one(A))
        <=> ( gcd_gcd(A,A2,B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_6611_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),gcd_gcd(int,M,Na))
    <=> ( ( M != zero_zero(int) )
        | ( Na != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_6612_gcd__neg__numeral__1__int,axiom,
    ! [Na: num,X: int] : gcd_gcd(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)),X) = gcd_gcd(int,aa(num,int,numeral_numeral(int),Na),X) ).

% gcd_neg_numeral_1_int
tff(fact_6613_gcd__neg__numeral__2__int,axiom,
    ! [X: int,Na: num] : gcd_gcd(int,X,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = gcd_gcd(int,X,aa(num,int,numeral_numeral(int),Na)) ).

% gcd_neg_numeral_2_int
tff(fact_6614_gcd__0__int,axiom,
    ! [X: int] : gcd_gcd(int,X,zero_zero(int)) = aa(int,int,abs_abs(int),X) ).

% gcd_0_int
tff(fact_6615_gcd__0__left__int,axiom,
    ! [X: int] : gcd_gcd(int,zero_zero(int),X) = aa(int,int,abs_abs(int),X) ).

% gcd_0_left_int
tff(fact_6616_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( dvd_dvd(int,Y,X)
     => ( gcd_gcd(int,X,Y) = aa(int,int,abs_abs(int),Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_6617_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( dvd_dvd(int,X,Y)
     => ( gcd_gcd(int,X,Y) = aa(int,int,abs_abs(int),X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_6618_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xsa: list(C),A2: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F2),Xsa)),A2) = aa(A,A,foldr(C,A,comp(B,fun(A,A),C,G,F2),Xsa),A2) ).

% foldr_map
tff(fact_6619_foldr__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),X: B,Xsa: list(B)] : foldr(B,A,F2,aa(list(B),list(B),cons(B,X),Xsa)) = comp(A,A,A,aa(B,fun(A,A),F2,X),foldr(B,A,F2,Xsa)) ).

% foldr_Cons
tff(fact_6620_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_gcd(int,X,Y)) ).

% gcd_ge_0_int
tff(fact_6621_gcd__red__int,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,X,Y) = gcd_gcd(int,Y,modulo_modulo(int,X,Y)) ).

% gcd_red_int
tff(fact_6622_comp__cong,axiom,
    ! [C: $tType,B: $tType,D6: $tType,A: $tType,E4: $tType,F2: fun(B,A),G: fun(C,B),X: C,F6: fun(D6,A),G3: fun(E4,D6),X6: E4] :
      ( ( aa(B,A,F2,aa(C,B,G,X)) = aa(D6,A,F6,aa(E4,D6,G3,X6)) )
     => ( aa(C,A,comp(B,A,C,F2,G),X) = aa(E4,A,comp(D6,A,E4,F6,G3),X6) ) ) ).

% comp_cong
tff(fact_6623_gcd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A,C2: A] : gcd_gcd(A,B2,gcd_gcd(A,A2,C2)) = gcd_gcd(A,A2,gcd_gcd(A,B2,C2)) ) ).

% gcd.left_commute
tff(fact_6624_gcd_Ocommute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : gcd_gcd(A,A2,B2) = gcd_gcd(A,B2,A2) ) ).

% gcd.commute
tff(fact_6625_gcd_Oassoc,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] : gcd_gcd(A,gcd_gcd(A,A2,B2),C2) = gcd_gcd(A,A2,gcd_gcd(A,B2,C2)) ) ).

% gcd.assoc
tff(fact_6626_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,Na: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),Na)) = gcd_gcd(A,M,Na) ) ).

% gcd_add_mult
tff(fact_6627_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M: A,Na: A] : gcd_gcd(A,aa(A,A,minus_minus(A,M),Na),Na) = gcd_gcd(A,M,Na) ) ).

% gcd_diff1
tff(fact_6628_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Na: A,M: A] : gcd_gcd(A,aa(A,A,minus_minus(A,Na),M),Na) = gcd_gcd(A,M,Na) ) ).

% gcd_diff2
tff(fact_6629_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y)) = gcd_gcd(int,X,Y) ).

% bezout_int
tff(fact_6630_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,Na: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),gcd_gcd(int,M,Na)) = gcd_gcd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),Na)) ).

% gcd_mult_distrib_int
tff(fact_6631_gcd__idem__int,axiom,
    ! [X: int] : gcd_gcd(int,X,X) = aa(int,int,abs_abs(int),X) ).

% gcd_idem_int
tff(fact_6632_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K: A] : dvd_dvd(A,gcd_gcd(A,A2,B2),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_6633_gcd__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( dvd_dvd(A,A2,C2)
         => ( dvd_dvd(A,B2,D3)
           => dvd_dvd(A,gcd_gcd(A,A2,B2),gcd_gcd(A,C2,D3)) ) ) ) ).

% gcd_mono
tff(fact_6634_dvd__gcdD1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,gcd_gcd(A,B2,C2))
         => dvd_dvd(A,A2,B2) ) ) ).

% dvd_gcdD1
tff(fact_6635_dvd__gcdD2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,gcd_gcd(A,B2,C2))
         => dvd_dvd(A,A2,C2) ) ) ).

% dvd_gcdD2
tff(fact_6636_gcd__dvdI1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,A2,C2)
         => dvd_dvd(A,gcd_gcd(A,A2,B2),C2) ) ) ).

% gcd_dvdI1
tff(fact_6637_gcd__dvdI2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( dvd_dvd(A,B2,C2)
         => dvd_dvd(A,gcd_gcd(A,A2,B2),C2) ) ) ).

% gcd_dvdI2
tff(fact_6638_gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => dvd_dvd(A,C2,gcd_gcd(A,A2,B2)) ) ) ) ).

% gcd_greatest
tff(fact_6639_gcd__dvd__antisym,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( dvd_dvd(A,gcd_gcd(A,A2,B2),gcd_gcd(A,C2,D3))
         => ( dvd_dvd(A,gcd_gcd(A,C2,D3),gcd_gcd(A,A2,B2))
           => ( gcd_gcd(A,A2,B2) = gcd_gcd(A,C2,D3) ) ) ) ) ).

% gcd_dvd_antisym
tff(fact_6640_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X4)),aa(B,A,H,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(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_6641_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6642_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6643_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_div_unit2
tff(fact_6644_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2),C2) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_div_unit1
tff(fact_6645_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F6: fun(A,B),A7: set(A),A13: set(B),F2: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F6,A7,A13)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),A3)),A7)
       => ( bij_betw(C,A,F2,A3,A7)
        <=> bij_betw(C,B,comp(A,B,C,F6,F2),A3,A13) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_6646_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,A2,B2)),B2) ) ).

% gcd_le2_int
tff(fact_6647_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,A2,B2)),A2) ) ).

% gcd_le1_int
tff(fact_6648_uminus__sum__list__map,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),Xsa: list(B)] : aa(A,A,uminus_uminus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xsa))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,comp(A,A,B,uminus_uminus(A),F2)),Xsa)) ) ).

% uminus_sum_list_map
tff(fact_6649_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,gcd_gcd(int,X,Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,gcd_gcd(int,X,aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),X),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),X),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,gcd_gcd(int,X,Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_6650_gcd__unique__int,axiom,
    ! [D3: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D3)
        & dvd_dvd(int,D3,A2)
        & dvd_dvd(int,D3,B2)
        & ! [E3: int] :
            ( ( dvd_dvd(int,E3,A2)
              & dvd_dvd(int,E3,B2) )
           => dvd_dvd(int,E3,D3) ) )
    <=> ( D3 = gcd_gcd(int,A2,B2) ) ) ).

% gcd_unique_int
tff(fact_6651_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Y)
     => ( gcd_gcd(int,X,Y) = gcd_gcd(int,Y,modulo_modulo(int,X,Y)) ) ) ).

% gcd_non_0_int
tff(fact_6652_gcd__code__int,axiom,
    ! [K: int,L: int] :
      gcd_gcd(int,K,L) = aa(int,int,abs_abs(int),
        $ite(L = zero_zero(int),K,gcd_gcd(int,L,modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L))))) ).

% gcd_code_int
tff(fact_6653_case__prod__comp,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F2: fun(D6,fun(C,A)),G: fun(B,D6),X: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),comp(D6,fun(C,A),B,F2,G)),X) = aa(C,A,aa(D6,fun(C,A),F2,aa(B,D6,G,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_6654_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)
         => ( ! [X4: A,Y3: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => ( aa(set(A),$o,member(A,Y3),A3)
                 => ( ( X4 != Y3 )
                   => ( ( aa(A,B,H,X4) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X4)) = 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),image(A,B,H),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),comp(B,C,A,G,H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_6655_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,comp(A,fun(A,A),B,plus_plus(A),G),zero_zero(A),A3) ) ).

% sum.eq_fold
tff(fact_6656_gcd__is__Max__divisors__int,axiom,
    ! [Na: int,M: int] :
      ( ( Na != zero_zero(int) )
     => ( gcd_gcd(int,M,Na) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ui(int,fun(int,fun(int,$o)),Na),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6657_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] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(B,$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),image(A,C,F2),I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),comp(C,B,A,G,F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_6658_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,minus_minus(nat,X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,minus_minus(nat,Xa),aa(nat,nat,suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_6659_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,minus_minus(nat,K),M)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_6660_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( gcd_gcd(nat,A2,B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_6661_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : gcd_gcd(nat,zero_zero(nat),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_6662_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = gcd_gcd(nat,A2,B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_6663_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : gcd_gcd(nat,A2,zero_zero(nat)) = A2 ).

% gcd_nat.right_neutral
tff(fact_6664_gcd__0__nat,axiom,
    ! [X: nat] : gcd_gcd(nat,X,zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_6665_gcd__0__left__nat,axiom,
    ! [X: nat] : gcd_gcd(nat,zero_zero(nat),X) = X ).

% gcd_0_left_nat
tff(fact_6666_gcd__1__nat,axiom,
    ! [M: nat] : gcd_gcd(nat,M,one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_6667_gcd__nat_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( gcd_gcd(nat,A2,B2) = A2 ) ) ).

% gcd_nat.absorb1
tff(fact_6668_gcd__nat_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( dvd_dvd(nat,B2,A2)
     => ( gcd_gcd(nat,A2,B2) = B2 ) ) ).

% gcd_nat.absorb2
tff(fact_6669_gcd__nat_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( dvd_dvd(nat,A2,gcd_gcd(nat,B2,C2))
    <=> ( dvd_dvd(nat,A2,B2)
        & dvd_dvd(nat,A2,C2) ) ) ).

% gcd_nat.bounded_iff
tff(fact_6670_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( dvd_dvd(nat,X,Y)
     => ( gcd_gcd(nat,X,Y) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_6671_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( dvd_dvd(nat,Y,X)
     => ( gcd_gcd(nat,X,Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_6672_gcd__Suc__0,axiom,
    ! [M: nat] : gcd_gcd(nat,M,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_6673_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),gcd_gcd(nat,M,Na))
    <=> ( ( M != zero_zero(nat) )
        | ( Na != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6674_gcd__int__int__eq,axiom,
    ! [M: nat,Na: nat] : gcd_gcd(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,M,Na)) ).

% gcd_int_int_eq
tff(fact_6675_size__list__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xsa: list(B)] : aa(list(A),nat,size_list(A,F2),aa(list(B),list(A),map(B,A,G),Xsa)) = aa(list(B),nat,size_list(B,comp(A,nat,B,F2,G)),Xsa) ).

% size_list_map
tff(fact_6676_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xsa: 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),Xsa))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,comp(A,$o,B,P,F2)),Xsa)) ).

% length_filter_map
tff(fact_6677_gcd__nat__abs__right__eq,axiom,
    ! [Na: nat,K: int] : gcd_gcd(nat,Na,aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,gcd_gcd(int,aa(nat,int,semiring_1_of_nat(int),Na),K)) ).

% gcd_nat_abs_right_eq
tff(fact_6678_gcd__nat__abs__left__eq,axiom,
    ! [K: int,Na: nat] : gcd_gcd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Na) = aa(int,nat,nat2,gcd_gcd(int,K,aa(nat,int,semiring_1_of_nat(int),Na))) ).

% gcd_nat_abs_left_eq
tff(fact_6679_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_uj(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_6680_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uk(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_6681_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_uj(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_6682_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uk(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_6683_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F2),Xsa)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,comp(A,$o,B,P,F2)),Xsa)) ).

% filter_map
tff(fact_6684_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : comp(list(B),nat,list(A),size_list(B,F2),map(A,B,G)) = size_list(A,comp(B,nat,A,F2,G)) ).

% list.size_gen_o_map
tff(fact_6685_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : takeWhile(A,P,aa(list(B),list(A),map(B,A,F2),Xsa)) = aa(list(B),list(A),map(B,A,F2),takeWhile(B,comp(A,$o,B,P,F2),Xsa)) ).

% takeWhile_map
tff(fact_6686_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),A2) ) ).

% gcd_le1_nat
tff(fact_6687_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),B2) ) ).

% gcd_le2_nat
tff(fact_6688_gcd__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : gcd_gcd(code_integer,code_integer_of_int(Xa),code_integer_of_int(X)) = code_integer_of_int(gcd_gcd(int,Xa,X)) ).

% gcd_integer.abs_eq
tff(fact_6689_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : gcd_gcd(nat,X,Y) = gcd_gcd(nat,Y,modulo_modulo(nat,X,Y)) ).

% gcd_red_nat
tff(fact_6690_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( gcd_gcd(nat,X,Y) = gcd_gcd(nat,Y,modulo_modulo(nat,X,Y)) ) ) ).

% gcd_non_0_nat
tff(fact_6691_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      gcd_gcd(nat,X,Y) = $ite(Y = zero_zero(nat),X,gcd_gcd(nat,Y,modulo_modulo(nat,X,Y))) ).

% gcd_nat.simps
tff(fact_6692_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( gcd_gcd(nat,X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero(nat),X,gcd_gcd(nat,Xa,modulo_modulo(nat,X,Xa))) ) ) ).

% gcd_nat.elims
tff(fact_6693_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,Na),M),Na) = gcd_gcd(nat,M,Na) ) ) ).

% gcd_diff2_nat
tff(fact_6694_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) = gcd_gcd(nat,M,Na) ) ) ).

% gcd_diff1_nat
tff(fact_6695_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),gcd_gcd(nat,M,Na)) = gcd_gcd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% gcd_mult_distrib_nat
tff(fact_6696_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(gcd_gcd(code_integer,X,Xa)) = gcd_gcd(int,code_int_of_integer(X),code_int_of_integer(Xa)) ).

% gcd_integer.rep_eq
tff(fact_6697_gcd__nat_Omono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D3: nat] :
      ( dvd_dvd(nat,A2,C2)
     => ( dvd_dvd(nat,B2,D3)
       => dvd_dvd(nat,gcd_gcd(nat,A2,B2),gcd_gcd(nat,C2,D3)) ) ) ).

% gcd_nat.mono
tff(fact_6698_gcd__nat_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( A2 = gcd_gcd(nat,A2,B2) ) ) ).

% gcd_nat.orderE
tff(fact_6699_gcd__nat_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = gcd_gcd(nat,A2,B2) )
     => dvd_dvd(nat,A2,B2) ) ).

% gcd_nat.orderI
tff(fact_6700_gcd__nat_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => ( gcd_gcd(nat,A2,B2) = A2 ) ) ).

% gcd_nat.absorb3
tff(fact_6701_gcd__nat_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd(nat,B2,A2)
        & ( B2 != A2 ) )
     => ( gcd_gcd(nat,A2,B2) = B2 ) ) ).

% gcd_nat.absorb4
tff(fact_6702_gcd__nat_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( dvd_dvd(nat,A2,gcd_gcd(nat,B2,C2))
     => ~ ( dvd_dvd(nat,A2,B2)
         => ~ dvd_dvd(nat,A2,C2) ) ) ).

% gcd_nat.boundedE
tff(fact_6703_gcd__nat_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( dvd_dvd(nat,A2,C2)
       => dvd_dvd(nat,A2,gcd_gcd(nat,B2,C2)) ) ) ).

% gcd_nat.boundedI
tff(fact_6704_gcd__nat_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
    <=> ( A2 = gcd_gcd(nat,A2,B2) ) ) ).

% gcd_nat.order_iff
tff(fact_6705_gcd__nat_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : dvd_dvd(nat,gcd_gcd(nat,A2,B2),A2) ).

% gcd_nat.cobounded1
tff(fact_6706_gcd__nat_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : dvd_dvd(nat,gcd_gcd(nat,A2,B2),B2) ).

% gcd_nat.cobounded2
tff(fact_6707_gcd__nat_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
    <=> ( gcd_gcd(nat,A2,B2) = A2 ) ) ).

% gcd_nat.absorb_iff1
tff(fact_6708_gcd__nat_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( dvd_dvd(nat,B2,A2)
    <=> ( gcd_gcd(nat,A2,B2) = B2 ) ) ).

% gcd_nat.absorb_iff2
tff(fact_6709_gcd__nat_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,C2)
     => dvd_dvd(nat,gcd_gcd(nat,A2,B2),C2) ) ).

% gcd_nat.coboundedI1
tff(fact_6710_gcd__nat_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( dvd_dvd(nat,B2,C2)
     => dvd_dvd(nat,gcd_gcd(nat,A2,B2),C2) ) ).

% gcd_nat.coboundedI2
tff(fact_6711_gcd__nat_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd(nat,A2,gcd_gcd(nat,B2,C2))
        & ( A2 != gcd_gcd(nat,B2,C2) ) )
     => ~ ( ( dvd_dvd(nat,A2,B2)
            & ( A2 != B2 ) )
         => ~ ( dvd_dvd(nat,A2,C2)
              & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_6712_gcd__nat_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
    <=> ( ( A2 = gcd_gcd(nat,A2,B2) )
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_6713_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,C2)
        & ( A2 != C2 ) )
     => ( dvd_dvd(nat,gcd_gcd(nat,A2,B2),C2)
        & ( gcd_gcd(nat,A2,B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_6714_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( dvd_dvd(nat,B2,C2)
        & ( B2 != C2 ) )
     => ( dvd_dvd(nat,gcd_gcd(nat,A2,B2),C2)
        & ( gcd_gcd(nat,A2,B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_6715_gcd__unique__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,D3,A2)
        & dvd_dvd(nat,D3,B2)
        & ! [E3: nat] :
            ( ( dvd_dvd(nat,E3,A2)
              & dvd_dvd(nat,E3,B2) )
           => dvd_dvd(nat,E3,D3) ) )
    <=> ( D3 = gcd_gcd(nat,A2,B2) ) ) ).

% gcd_unique_nat
tff(fact_6716_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F2: fun(B,A),X: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,member(B,X),S2)
       => ( aa(set(B),$o,member(B,Y),S2)
         => ( comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X)) = comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_6717_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X4: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),gcd_gcd(nat,A2,B2)) ) ).

% bezout_nat
tff(fact_6718_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X4: nat,Y3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = gcd_gcd(nat,A2,B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = gcd_gcd(nat,A2,B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6719_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : bit_se5641148757651400278ts_bit(A,bit_se4197421643247451524op_bit(A,Na,A2)) = comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2),aa(nat,fun(nat,nat),plus_plus(nat),Na)) ) ).

% bit_drop_bit_eq
tff(fact_6720_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ul(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6721_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      gcd_gcd(code_integer,K,L) = aa(code_integer,code_integer,abs_abs(code_integer),
        $ite(L = zero_zero(code_integer),K,gcd_gcd(code_integer,L,modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))))) ).

% gcd_code_integer
tff(fact_6722_num__of__nat__code,axiom,
    num_of_nat = comp(code_integer,num,nat,code_num_of_integer,semiring_1_of_nat(code_integer)) ).

% num_of_nat_code
tff(fact_6723_gcd__int__def,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,X,Y) = aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),X)),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ).

% gcd_int_def
tff(fact_6724_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( gcd_gcd(nat,M,Na) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_um(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6725_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_6726_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6727_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6728_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6729_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_6730_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_6731_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aTP_Lamp_no(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_6732_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aTP_Lamp_no(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_6733_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_no(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6734_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_no(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6735_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(int,A,nat,G,semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_6736_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : groups7121269368397514597t_prod(int,A,G,set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_6737_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_6738_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(int,A,nat,G,semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_6739_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xsa: list(B)] : map_filter(B,A,F2,Xsa) = aa(list(B),list(A),map(B,A,comp(option(A),A,B,the2(A),F2)),aa(list(B),list(B),filter2(B,aTP_Lamp_un(fun(B,option(A)),fun(B,$o),F2)),Xsa)) ).

% map_filter_def
tff(fact_6740_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6741_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : groups7121269368397514597t_prod(int,A,G,set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_6742_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,X,Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).

% bezw_aux
tff(fact_6743_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,minus_minus(nat,X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,minus_minus(nat,Xa),aa(nat,nat,suc,X)))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_6744_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( gcd_gcd(nat,X,Xa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero(nat),X,gcd_gcd(nat,Xa,modulo_modulo(nat,X,Xa))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_6745_gcd__idem__nat,axiom,
    ! [X: nat] : gcd_gcd(nat,X,X) = X ).

% gcd_idem_nat
tff(fact_6746_gcd__nat_Oright__idem,axiom,
    ! [A2: nat,B2: nat] : gcd_gcd(nat,gcd_gcd(nat,A2,B2),B2) = gcd_gcd(nat,A2,B2) ).

% gcd_nat.right_idem
tff(fact_6747_gcd__nat_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] : gcd_gcd(nat,A2,gcd_gcd(nat,A2,B2)) = gcd_gcd(nat,A2,B2) ).

% gcd_nat.left_idem
tff(fact_6748_gcd__nat_Oidem,axiom,
    ! [A2: nat] : gcd_gcd(nat,A2,A2) = A2 ).

% gcd_nat.idem
tff(fact_6749_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_uo(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_6750_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_up(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_6751_refl__ge__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X4: A] : aa(A,$o,aa(A,fun(A,$o),R2,X4),X4)
     => aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R2) ) ).

% refl_ge_eq
tff(fact_6752_ge__eq__refl,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),X: A] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R2)
     => aa(A,$o,aa(A,fun(A,$o),R2,X),X) ) ).

% ge_eq_refl
tff(fact_6753_Code__Numeral_Onegative__def,axiom,
    code_negative = comp(code_integer,code_integer,num,uminus_uminus(code_integer),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_6754_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp(int,int,num,uminus_uminus(int),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_6755_empty__natural,axiom,
    ! [C: $tType,B: $tType,D6: $tType,A: $tType,F2: fun(A,C),G: fun(D6,B)] : comp(C,set(B),A,aTP_Lamp_uq(C,set(B)),F2) = comp(set(D6),set(B),A,image(D6,B,G),aTP_Lamp_ur(A,set(D6))) ).

% empty_natural
tff(fact_6756_finite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [R: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S2)))
          & ! [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),aa(set(nat),nat,finite_card(nat),S2))
             => aa(set(nat),$o,member(nat,aa(nat,nat,R,N8)),S2) ) ) ) ).

% finite_enumerate
tff(fact_6757_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A3: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( aa(set(A),$o,member(A,X),A3)
           => ( aa(set(A),$o,member(A,Y),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_6758_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A),R3: A,S: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( aa(set(A),$o,member(A,R3),A3)
           => ( aa(set(A),$o,member(A,S),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_6759_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [R: A,S3: A] :
              ( aa(set(A),$o,member(A,R),A3)
             => ( aa(set(A),$o,member(A,S3),A3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R),S3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R)),aa(A,B,F2,S3)) ) ) )
         => strict_mono_on(A,B,F2,A3) ) ) ).

% strict_mono_onI
tff(fact_6760_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F2,A3)
        <=> ! [R5: A,S7: A] :
              ( ( aa(set(A),$o,member(A,R5),A3)
                & aa(set(A),$o,member(A,S7),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S7) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S7)) ) ) ) ).

% strict_mono_on_def
tff(fact_6761_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),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% conj_subset_def
tff(fact_6762_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_us(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q),G),X3)
    <=> ( aa(A,$o,comp(B,$o,A,P,G),X3)
        & aa(A,$o,comp(B,$o,A,Q,G),X3) ) ) ).

% conj_comp_iff
tff(fact_6763_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A3)
     => ( ! [X4: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X4),A3)
           => aa(set(A),$o,finite_finite2(A),X4) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_ut(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_uu(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).

% card_UNION
tff(fact_6764_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),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),finite_Fpow(B,A3))),finite_Fpow(A,B3)) ) ).

% image_Fpow_mono
tff(fact_6765_cSup__lessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A)
        & no_bot(A) )
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),X)) = X ) ).

% cSup_lessThan
tff(fact_6766_cSup__atMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),X)) = X ) ).

% cSup_atMost
tff(fact_6767_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_6768_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_6769_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cSup_singleton
tff(fact_6770_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( complete_Inf_Inf(A,set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_6771_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( complete_Inf_Inf(A,set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_6772_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : complete_Inf_Inf(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cInf_singleton
tff(fact_6773_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_6774_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_6775_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( complete_Inf_Inf(A,set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_6776_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( complete_Inf_Inf(A,set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_6777_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),image(A,B,aTP_Lamp_uv(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cSUP_const
tff(fact_6778_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_uv(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cINF_const
tff(fact_6779_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] :
            ( aa(set(A),$o,member(A,A4),A3)
           => aa(set(B),$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)),image(A,set(B),B3),A3))) ) ) ).

% finite_UN_I
tff(fact_6780_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),I5)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
     => aa(set(B),$o,finite_finite2(B),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ).

% finite_INT
tff(fact_6781_Union__natural,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B)),image(set(A),set(B),image(A,B,F2))) = comp(set(A),set(B),set(set(A)),image(A,B,F2),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_6782_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A3: C] :
      ( aa(set(A),$o,member(A,X),aa(C,set(A),comp(B,set(A),C,comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A)),Gset),Gmap),A3))
     => aa(set(A),$o,member(A,X),aa(C,set(A),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A)),comp(B,set(set(A)),C,Gset,Gmap)),A3)) ) ).

% in_Union_o_assoc
tff(fact_6783_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),X: C,X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),aa(C,set(B),G,X)))),X5)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,X)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_uw(fun(B,set(A)),fun(set(A),fun(B,$o)),F2),X5))) ) ).

% UN_image_subset
tff(fact_6784_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),X: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,X),X5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),complete_Inf_Inf(A,X5)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_6785_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),X: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,X),X5)
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),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_6786_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),complete_Inf_Inf(A,X5)),Z2)
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z2) ) ) ) ) ).

% cInf_lessD
tff(fact_6787_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X4) ) ) ) ) ).

% less_cSupE
tff(fact_6788_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X5))
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X4) ) ) ) ) ).

% less_cSupD
tff(fact_6789_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_uy(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3)),A3)) ) ).

% SUP_inf_distrib2
tff(fact_6790_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uz(A,fun(fun(B,A),fun(B,A)),A2),F2)),B3)) ) ).

% inf_SUP
tff(fact_6791_Sup__inf,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) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_va(A,fun(A,A),A2)),B3)) ) ).

% Sup_inf
tff(fact_6792_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B3: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),B3))),A2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_vb(fun(B,A),fun(A,fun(B,A)),F2),A2)),B3)) ) ).

% SUP_inf
tff(fact_6793_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3))),complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,G),B3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_vd(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3)),A3)) ) ).

% INF_sup_distrib2
tff(fact_6794_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),B3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ve(A,fun(fun(B,A),fun(B,A)),A2),F2)),B3)) ) ).

% sup_INF
tff(fact_6795_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,B3)),A2) = complete_Inf_Inf(A,aa(set(A),set(A),image(A,A,aTP_Lamp_vf(A,fun(A,A),A2)),B3)) ) ).

% Inf_sup
tff(fact_6796_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B3: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),B3))),A2) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_vg(fun(B,A),fun(A,fun(B,A)),F2),A2)),B3)) ) ).

% INF_sup
tff(fact_6797_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,X),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% le_cSup_finite
tff(fact_6798_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z2),X5)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X4) )
           => ( complete_Inf_Inf(A,X5) = Z2 ) ) ) ) ).

% cInf_eq_minimum
tff(fact_6799_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4) )
         => ( ! [Y3: A] :
                ( ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
           => ( complete_Inf_Inf(A,X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_6800_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z2),X5)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z2) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = Z2 ) ) ) ) ).

% cSup_eq_maximum
tff(fact_6801_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) )
         => ( ! [Y3: A] :
                ( ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_6802_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,X),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,X5)),X) ) ) ) ).

% cInf_le_finite
tff(fact_6803_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X4) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),complete_Inf_Inf(A,X5)) ) ) ) ).

% cInf_greatest
tff(fact_6804_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4) )
           => ( ! [Y3: A] :
                  ( ! [X3: A] :
                      ( aa(set(A),$o,member(A,X3),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
             => ( complete_Inf_Inf(A,X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_6805_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z2) ) ) ) ).

% cSup_least
tff(fact_6806_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) )
           => ( ! [Y3: A] :
                  ( ! [X3: A] :
                      ( aa(set(A),$o,member(A,X3),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6807_card__Union__le__sum__card,axiom,
    ! [A: $tType,U3: set(set(A))] : aa(nat,$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)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3)) ).

% card_Union_le_sum_card
tff(fact_6808_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),M: B,F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F2,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ).

% cINF_greatest
tff(fact_6809_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),M7: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),M7) )
           => 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),image(A,B,F2),A3))),M7) ) ) ) ).

% cSUP_least
tff(fact_6810_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),complete_Inf_Inf(A,X5))
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_6811_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] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6812_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),complete_Inf_Inf(A,S2))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_6813_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),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_6814_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X4: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X4),B3)
             => aa(set(A),$o,finite_finite2(A),X4) )
         => ( ! [A14: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),A14),B3)
               => ! [A24: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),A24),B3)
                   => ( ( A14 != A24 )
                     => ! [X4: A] :
                          ( aa(set(A),$o,member(A,X4),A14)
                         => ( aa(set(A),$o,member(A,X4),A24)
                           => ( aa(A,B,G,X4) = 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),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_6815_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_6816_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_6817_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) = complete_Inf_Inf(A,A3) ) ) ) ) ).

% Min_Inf
tff(fact_6818_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( complete_Inf_Inf(A,X5) = aa(set(A),A,lattic643756798350308766er_Min(A),X5) ) ) ) ) ).

% cInf_eq_Min
tff(fact_6819_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U3: set(set(A))] :
      ( ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),U3)
         => aa(set(A),$o,finite_finite2(A),X4) )
     => aa(nat,$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)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3)) ) ).

% card_Union_le_sum_card_weak
tff(fact_6820_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)) )
           => ( complete_Inf_Inf(A,X5) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X5) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_6821_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) = complete_Inf_Inf(A,A3) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6822_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),A2: B,B3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,A2)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),B3))) ).

% Union_image_insert
tff(fact_6823_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_6824_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_6825_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M7: fun(nat,set(A)),K: nat,Na: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_vh(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M7),K)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M7),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)))) ).

% UN_le_add_shift_strict
tff(fact_6826_UN__le__add__shift,axiom,
    ! [A: $tType,M7: fun(nat,set(A)),K: nat,Na: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_vh(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M7),K)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M7),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)))) ).

% UN_le_add_shift
tff(fact_6827_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X4),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,complete_Inf_Inf(A,S2)),L))),E2) ) ) ) ).

% cInf_asclose
tff(fact_6828_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X4),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S2)),L))),E2) ) ) ) ).

% cSup_asclose
tff(fact_6829_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_6830_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S2)) = $ite(S2 = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S2))) ) ) ) ).

% Sup_insert_finite
tff(fact_6831_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_6832_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_vi(set(A),fun(set(A),$o),A3)) ).

% Fpow_def
tff(fact_6833_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B14: set(set(A))] :
      ( aa(set(A),$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)),B14))
       => ~ ! [F8: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F8)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F8),B14)
               => ~ aa(set(A),$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)),F8)) ) ) ) ) ).

% finite_subset_Union
tff(fact_6834_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B3: set(A),X: B,A3: set(product_prod(B,A))] :
      ( aa(set(A),$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))),image(A,set(product_prod(B,A)),aTP_Lamp_vj(B,fun(A,set(product_prod(B,A))),X)),B3))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_vk(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A3,B3) ) ) ).

% fold_union_pair
tff(fact_6835_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
     => ( ! [X4: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X4),C3)
           => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),X4)) )
       => ( ! [X4: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X4),C3)
             => ! [Xa4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                 => ( ( X4 != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),Xa4) = bot_bot(set(A)) ) ) ) )
         => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))) ) ) ) ).

% dvd_partition
tff(fact_6836_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)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X4)) )
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I5)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I5)
                     => ( ( X4 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X4)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),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)),image(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_vl(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_6837_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)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X4)) )
           => ( ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I5)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I5)
                     => ( ( X4 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X4)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( groups7121269368397514597t_prod(B,C,G,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vm(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_6838_UN__le__eq__Un0,axiom,
    ! [A: $tType,M7: fun(nat,set(A)),Na: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M7),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M7),set_or1337092689740270186AtMost(nat,one_one(nat),Na)))),aa(nat,set(A),M7,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_6839_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)),image(A,set(B),A3),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_vn(fun(A,set(B)),fun(A,nat),A3)),I5)) ) ).

% card_UN_le
tff(fact_6840_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),I5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X4)) )
       => ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),I5)
             => ! [Xa4: A] :
                  ( aa(set(A),$o,member(A,Xa4),I5)
                 => ( ( X4 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X4)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_vn(fun(A,set(B)),fun(A,nat),A3)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_6841_Fpow__Pow__finite,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A3)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_6842_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_vo(fun(B,set(A)),fun(B,set(A)),B3)),A3)) ).

% Compl_INT
tff(fact_6843_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_vo(fun(B,set(A)),fun(B,set(A)),B3)),A3)) ).

% Compl_UN
tff(fact_6844_Sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(fun(B,A)),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),A3),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_vp(B,fun(fun(B,A),A),X)),A3)) ) ).

% Sup_apply
tff(fact_6845_Inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(fun(B,A)),X: B] : aa(B,A,complete_Inf_Inf(fun(B,A),A3),X) = complete_Inf_Inf(A,aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_vq(B,fun(fun(B,A),A),X)),A3)) ) ).

% Inf_apply
tff(fact_6846_SUP__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_vr(set(A),fun(fun(A,$o),$o)) = bex(A) ).

% SUP_bool_eq
tff(fact_6847_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_6848_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F2: fun(C,fun(B,A)),A3: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F2),A3)),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_vs(fun(C,fun(B,A)),fun(B,fun(C,A)),F2),X)),A3)) ) ).

% SUP_apply
tff(fact_6849_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_vt(A,A)),A3)) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ).

% SUP_identity_eq
tff(fact_6850_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F2: fun(C,fun(B,A)),A3: set(C),X: B] : aa(B,A,complete_Inf_Inf(fun(B,A),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F2),A3)),X) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_vu(fun(C,fun(B,A)),fun(B,fun(C,A)),F2),X)),A3)) ) ).

% INF_apply
tff(fact_6851_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(A)] : complete_Inf_Inf(A,aa(set(A),set(A),image(A,A,aTP_Lamp_vv(A,A)),A3)) = complete_Inf_Inf(A,A3) ) ).

% INF_identity_eq
tff(fact_6852_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
          & aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X2)) ) ) ).

% UN_iff
tff(fact_6853_UN__I,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A2))
       => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ) ).

% UN_I
tff(fact_6854_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X2)) ) ) ).

% INT_iff
tff(fact_6855_INT__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B2: B,B3: fun(A,set(B))] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(B),$o,member(B,B2),aa(A,set(B),B3,X4)) )
     => aa(set(B),$o,member(B,B2),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ).

% INT_I
tff(fact_6856_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( complete_Inf_Inf(A,A3) = bot_bot(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X2) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_6857_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A2: A,A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ).

% Sup_insert
tff(fact_6858_Inf__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A2: A,A3: set(A)] : complete_Inf_Inf(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),complete_Inf_Inf(A,A3)) ) ).

% Inf_insert
tff(fact_6859_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),image(B,A,aTP_Lamp_vw(B,A)),A3)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_6860_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B3),A3)) = bot_bot(A) )
        <=> ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A3)
             => ( aa(B,A,B3,X2) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_6861_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B3),A3)) )
        <=> ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A3)
             => ( aa(B,A,B3,X2) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_6862_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),image(A,B,aTP_Lamp_vx(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% SUP_const
tff(fact_6863_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_vx(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% INF_const
tff(fact_6864_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)),image(B,set(A),aTP_Lamp_vy(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),C2) ).

% UN_constant
tff(fact_6865_UN__Un,axiom,
    ! [A: $tType,B: $tType,M7: fun(B,set(A)),A3: set(B),B3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M7),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(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M7),A3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M7),B3))) ).

% UN_Un
tff(fact_6866_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3)) = bot_bot(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa2)),X2) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_6867_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_vz(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),C3)))) ).

% UN_simps(1)
tff(fact_6868_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)),image(A,set(A),aTP_Lamp_wa(A,set(A))),A3)) = A3 ).

% UN_singleton
tff(fact_6869_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = 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)),image(B,set(A),B3),C3)))) ).

% UN_simps(3)
tff(fact_6870_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: set(A),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = 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)),image(B,set(A),A3),C3))),B3)) ).

% UN_simps(2)
tff(fact_6871_UN__insert,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A2: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B3,A2)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3))) ).

% UN_insert
tff(fact_6872_INT__insert,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A2: B,A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B3,A2)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3))) ).

% INT_insert
tff(fact_6873_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),S2),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),complete_Inf_Inf(set(product_prod(A,B)),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Inf_INT_eq2
tff(fact_6874_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S2),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Sup_SUP_eq2
tff(fact_6875_Inf__int__def,axiom,
    ! [X5: set(int)] : complete_Inf_Inf(int,X5) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),aa(set(int),set(int),image(int,int,uminus_uminus(int)),X5))) ).

% Inf_int_def
tff(fact_6876_Inf__real__def,axiom,
    ! [X5: set(real)] : complete_Inf_Inf(real,X5) = aa(real,real,uminus_uminus(real),aa(set(real),real,complete_Sup_Sup(real),aa(set(real),set(real),image(real,real,uminus_uminus(real)),X5))) ).

% Inf_real_def
tff(fact_6877_Inf__nat__def1,axiom,
    ! [K6: set(nat)] :
      ( ( K6 != bot_bot(set(nat)) )
     => aa(set(nat),$o,member(nat,complete_Inf_Inf(nat,K6)),K6) ) ).

% Inf_nat_def1
tff(fact_6878_Inf__nat__def,axiom,
    ! [X5: set(nat)] : complete_Inf_Inf(nat,X5) = ord_Least(nat,aTP_Lamp_wd(set(nat),fun(nat,$o),X5)) ).

% Inf_nat_def
tff(fact_6879_SUP__Sup__eq,axiom,
    ! [A: $tType,S2: set(set(A)),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o))),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2)) ) ).

% SUP_Sup_eq
tff(fact_6880_INF__Int__eq,axiom,
    ! [A: $tType,S2: set(set(A)),X3: A] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o))),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),complete_Inf_Inf(set(A),S2)) ) ).

% INF_Int_eq
tff(fact_6881_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S2: set(B),X3: A] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_we(fun(B,set(A)),fun(B,fun(A,$o)),R3)),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),R3),S2))) ) ).

% INF_INT_eq
tff(fact_6882_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S2: set(B),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_we(fun(B,set(A)),fun(B,fun(A,$o)),R3)),S2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),R3),S2))) ) ).

% SUP_UN_eq
tff(fact_6883_Inf__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : complete_Inf_Inf(set(A),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_wf(set(set(A)),fun(A,$o),A3)) ).

% Inf_set_def
tff(fact_6884_Sup__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_wg(set(set(A)),fun(A,$o),A3)) ).

% Sup_set_def
tff(fact_6885_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S2: set(C),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_wh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R3)),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R3),S2))) ) ).

% SUP_UN_eq2
tff(fact_6886_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S2: set(C),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_wh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R3)),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),complete_Inf_Inf(set(product_prod(A,B)),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R3),S2))) ) ).

% INF_INT_eq2
tff(fact_6887_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),complete_Inf_Inf(set(product_prod(A,B)),S2)) ) ).

% INF_Int_eq2
tff(fact_6888_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S2)) ) ).

% SUP_Sup_eq2
tff(fact_6889_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_6890_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A3: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image(A,A,aTP_Lamp_ab(A,A)),A3)) = aa(set(A),A,Sup,A3) ).

% Sup.SUP_identity_eq
tff(fact_6891_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A3: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image(A,A,aTP_Lamp_ab(A,A)),A3)) = aa(set(A),A,Inf,A3) ).

% Inf.INF_identity_eq
tff(fact_6892_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A3: set(fun(A,B)),X3: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A3),X3) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_wi(A,fun(fun(A,B),B),X3)),A3)) ) ).

% Sup_fun_def
tff(fact_6893_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A3: set(fun(A,B)),X3: A] : aa(A,B,complete_Inf_Inf(fun(A,B),A3),X3) = complete_Inf_Inf(B,aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_wj(A,fun(fun(A,B),B),X3)),A3)) ) ).

% Inf_fun_def
tff(fact_6894_Union__eq,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_wk(set(set(A)),fun(A,$o),A3)) ).

% Union_eq
tff(fact_6895_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( aa(set(A),$o,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_6896_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2)
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B2) ) ) ) ).

% Sup_le_iff
tff(fact_6897_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Sup_upper
tff(fact_6898_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z2: A] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z2) ) ) ).

% Sup_least
tff(fact_6899_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ! [A4: A] :
              ( aa(set(A),$o,member(A,A4),A3)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),B3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ).

% Sup_mono
tff(fact_6900_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [Y3: A] :
              ( aa(set(A),$o,member(A,Y3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
         => ( ! [Y3: A] :
                ( ! [Z3: A] :
                    ( aa(set(A),$o,member(A,Z3),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = X ) ) ) ) ).

% Sup_eqI
tff(fact_6901_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
        <=> ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) ) ) ) ).

% less_Sup_iff
tff(fact_6902_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z2: A] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),complete_Inf_Inf(A,A3)) ) ) ).

% Inf_greatest
tff(fact_6903_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),complete_Inf_Inf(A,A3))
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X2) ) ) ) ).

% le_Inf_iff
tff(fact_6904_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( aa(set(A),$o,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),complete_Inf_Inf(A,A3)),V2) ) ) ) ).

% Inf_lower2
tff(fact_6905_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),X) ) ) ).

% Inf_lower
tff(fact_6906_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ! [B4: A] :
              ( aa(set(A),$o,member(A,B4),B3)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B3)) ) ) ).

% Inf_mono
tff(fact_6907_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),I2) )
         => ( ! [Y3: A] :
                ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),A3)
                   => aa(A,$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),X) )
           => ( complete_Inf_Inf(A,A3) = X ) ) ) ) ).

% Inf_eqI
tff(fact_6908_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),complete_Inf_Inf(A,S2)),A2)
        <=> ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) ) ) ) ).

% Inf_less_iff
tff(fact_6909_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),C3) )
     => 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)),C3) ) ).

% Union_least
tff(fact_6910_Union__upper,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( aa(set(set(A)),$o,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_6911_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),A3)
         => ? [Y2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),Y2),B3)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Y2) ) )
     => aa(set(A),$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_6912_Inter__lower,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( aa(set(set(A)),$o,member(set(A),B3),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_Inf_Inf(set(A),A3)),B3) ) ).

% Inter_lower
tff(fact_6913_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),X7) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),complete_Inf_Inf(set(A),A3)) ) ).

% Inter_greatest
tff(fact_6914_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_wl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F2),B3)),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_wn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F2),A3)),B3)) ) ).

% SUP_commute
tff(fact_6915_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_wo(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F2),B3)),A3)) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_wp(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F2),A3)),B3)) ) ).

% INF_commute
tff(fact_6916_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,set(A)),B3: fun(C,set(B)),A3: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),C3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B3),A3)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wq(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C3),B3)),A3)) ).

% UN_UN_flatten
tff(fact_6917_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))
     => ~ ! [X4: B] :
            ( aa(set(B),$o,member(B,X4),A3)
           => ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X4)) ) ) ).

% UN_E
tff(fact_6918_UN__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B3: fun(B,set(C)),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_wr(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B3)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B3),A3)))) ).

% UN_extend_simps(9)
tff(fact_6919_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))
     => ( ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B3,A2))
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INT_E
tff(fact_6920_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(set(A),$o,member(A,B2),aa(B,set(A),B3,A2)) ) ) ).

% INT_D
tff(fact_6921_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X)
             => ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X2) ) ) ) ) ).

% le_Sup_iff
tff(fact_6922_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),X)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y5)
             => ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_6923_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,I2)),aa(B,C,G,X3)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B3)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F2,X3)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F2),A3)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B3)) ) ) ) ) ).

% SUP_eq
tff(fact_6924_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V4: A] :
              ( aa(set(A),$o,member(A,V4),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V4) )
         => ( ( 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_6925_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),G: fun(B,C),F2: fun(A,C)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X3)),aa(A,C,F2,I2)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B3)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X3)),aa(B,C,G,J2)) ) )
           => ( complete_Inf_Inf(C,aa(set(A),set(C),image(A,C,F2),A3)) = complete_Inf_Inf(C,aa(set(B),set(C),image(B,C,G),B3)) ) ) ) ) ).

% INF_eq
tff(fact_6926_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_6927_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V4: A] :
              ( aa(set(A),$o,member(A,V4),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V4),U) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),U) ) ) ) ).

% Inf_less_eq
tff(fact_6928_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),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B3)) ) ) ).

% Inf_superset_mono
tff(fact_6929_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_6930_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)),complete_Inf_Inf(set(A),A3)),complete_Inf_Inf(set(A),B3)) ) ).

% Inter_anti_mono
tff(fact_6931_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B3) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_Inf_Inf(set(A),A3)),B3) ) ) ).

% Inter_subset
tff(fact_6932_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),X: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),X) )
         => ( ! [Y3: B] :
                ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),A3)
                   => aa(B,$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),X),Y3) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_6933_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [N: A] :
              ( aa(set(A),$o,member(A,N),A3)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N)),aa(B,C,G,X3)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F2),A3))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B3))) ) ) ).

% SUP_mono
tff(fact_6934_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,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),image(A,B,F2),A3))),U) ) ) ).

% SUP_least
tff(fact_6935_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% SUP_mono'
tff(fact_6936_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% SUP_upper
tff(fact_6937_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,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U)
        <=> ! [X2: B] :
              ( aa(set(B),$o,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_6938_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),U: B,F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ).

% SUP_upper2
tff(fact_6939_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),Y)
         => ( aa(set(B),$o,member(B,I),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_6940_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,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ? [X2: B] :
              ( aa(set(B),$o,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_6941_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),X: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(A,B,F2,I2)) )
         => ( ! [Y3: B] :
                ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),A3)
                   => aa(B,$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),X) )
           => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3)) = X ) ) ) ) ).

% INF_eqI
tff(fact_6942_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B3: set(A),A3: set(B),F2: fun(B,C),G: fun(A,C)] :
          ( ! [M4: A] :
              ( aa(set(A),$o,member(A,M4),B3)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F2,X3)),aa(A,C,G,M4)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),complete_Inf_Inf(C,aa(set(B),set(C),image(B,C,F2),A3))),complete_Inf_Inf(C,aa(set(A),set(C),image(A,C,G),B3))) ) ) ).

% INF_mono
tff(fact_6943_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,I)) ) ) ).

% INF_lower
tff(fact_6944_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% INF_mono'
tff(fact_6945_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F2: fun(A,B),U: B] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))),U) ) ) ) ).

% INF_lower2
tff(fact_6946_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,$o),ord_less_eq(A),U),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ! [X2: B] :
              ( aa(set(B),$o,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_6947_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),U: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,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),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% INF_greatest
tff(fact_6948_subset__Pow__Union,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A3),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ).

% subset_Pow_Union
tff(fact_6949_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A3: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3)))
         => ( aa(set(B),$o,member(B,I),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I)) ) ) ) ).

% less_INF_D
tff(fact_6950_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,$o),ord_less(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3))),A2)
        <=> ? [X2: B] :
              ( aa(set(B),$o,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_6951_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ws(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_6952_SUP__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,A3,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% SUP_absorb
tff(fact_6953_INF__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,A3,K)),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,A3),I5))) = complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% INF_absorb
tff(fact_6954_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3))),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,G),A3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wt(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ).

% INF_inf_distrib
tff(fact_6955_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),B3: fun(C,set(B)),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B3),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F2),B3)),A3)) ).

% image_UN
tff(fact_6956_UN__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,set(A)),F2: fun(B,C),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_wv(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B3),F2)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B3),aa(set(B),set(C),image(B,C,F2),A3))) ).

% UN_extend_simps(10)
tff(fact_6957_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)),image(B,set(A),aTP_Lamp_ww(B,set(A))),A3)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_6958_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)),image(B,set(A),B3),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_6959_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)) )
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => ( aa(B,set(A),B3,X2) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_6960_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)) = bot_bot(set(A)) )
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => ( aa(B,set(A),B3,X2) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_6961_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),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X4)),aa(A,set(B),G,X4)) )
       => 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)),image(A,set(B),F2),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),B3))) ) ) ).

% UN_mono
tff(fact_6962_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(A,set(B)),C3: set(B)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,X4)),C3) )
     => 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)),image(A,set(B),B3),A3))),C3) ) ).

% UN_least
tff(fact_6963_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,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)),image(A,set(B),B3),A3))) ) ).

% UN_upper
tff(fact_6964_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),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),B3)
    <=> ! [X2: B] :
          ( aa(set(B),$o,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_6965_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_wx(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B3)),A3)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ) ).

% UN_insert_distrib
tff(fact_6966_UN__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: 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_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wy(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) ).

% UN_extend_simps(5)
tff(fact_6967_UN__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: 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_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wz(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) ).

% UN_extend_simps(4)
tff(fact_6968_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wy(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A3)),I5)) ).

% Int_UN_distrib
tff(fact_6969_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B3: fun(C,set(A)),J4: set(C)] : 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)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B3),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_xb(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B3),J4)),I5)) ).

% Int_UN_distrib2
tff(fact_6970_INT__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,set(A)),F2: fun(B,C),A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_wv(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B3),F2)),A3)) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B3),aa(set(B),set(C),image(B,C,F2),A3))) ).

% INT_extend_simps(10)
tff(fact_6971_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B3: fun(B,set(A)),C3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) ).

% Un_Union_image
tff(fact_6972_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B3: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),I5))) ).

% UN_Un_distrib
tff(fact_6973_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A3,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% UN_absorb
tff(fact_6974_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_xd(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) ).

% UN_extend_simps(6)
tff(fact_6975_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B3),A3))),aa(A,set(B),B3,A2)) ) ).

% INT_lower
tff(fact_6976_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(B),B3: fun(A,set(B))] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),aa(A,set(B),B3,X4)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ).

% INT_greatest
tff(fact_6977_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),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X4)),aa(A,set(B),G,X4)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),F2),B3))),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),G),A3))) ) ) ).

% INT_anti_mono
tff(fact_6978_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),$o),ord_less_eq(set(A)),B3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))
    <=> ! [X2: B] :
          ( aa(set(B),$o,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_6979_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_vz(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3)) ).

% INT_extend_simps(5)
tff(fact_6980_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_wx(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B3)),A3)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ) ).

% INT_insert_distrib
tff(fact_6981_Collect__bex__eq,axiom,
    ! [A: $tType,B: $tType,A3: set(B),P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_xe(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),A3),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_xg(fun(A,fun(B,$o)),fun(B,set(A)),P)),A3)) ).

% Collect_bex_eq
tff(fact_6982_UNION__eq,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_xh(fun(B,set(A)),fun(set(B),fun(A,$o)),B3),A3)) ).

% UNION_eq
tff(fact_6983_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,K)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% INT_absorb
tff(fact_6984_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B3: fun(B,set(A)),I5: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xi(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),I5))) ).

% INT_Int_distrib
tff(fact_6985_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B3: fun(B,set(A)),C3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xi(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) ).

% Int_Inter_image
tff(fact_6986_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B3: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B3),J4))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_xk(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B3),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_6987_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wb(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A3)),I5)) ).

% Un_INT_distrib
tff(fact_6988_INT__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) ).

% INT_extend_simps(6)
tff(fact_6989_INT__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) ).

% INT_extend_simps(7)
tff(fact_6990_INT__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B3: fun(B,set(C)),A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_xl(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B3)),A3)) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B3),A3)))) ).

% INT_extend_simps(9)
tff(fact_6991_image__Union,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(set(B))] : aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),S2)) ).

% image_Union
tff(fact_6992_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : pow2(A,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3))) = complete_Inf_Inf(set(set(A)),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_xm(fun(B,set(A)),fun(B,set(set(A))),B3)),A3)) ).

% Pow_INT_eq
tff(fact_6993_Int__Union,axiom,
    ! [A: $tType,A3: set(A),B3: 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_Sup_Sup(set(A)),B3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B3)) ).

% Int_Union
tff(fact_6994_Int__Union2,axiom,
    ! [A: $tType,B3: 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)),B3)),A3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_xn(set(A),fun(set(A),set(A)),A3)),B3)) ).

% Int_Union2
tff(fact_6995_UN__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(set(B))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_xo(fun(B,set(A)),fun(set(B),set(A)),B3)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% UN_extend_simps(8)
tff(fact_6996_Un__Inter,axiom,
    ! [A: $tType,A3: set(A),B3: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),complete_Inf_Inf(set(A),B3)) = complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3)),B3)) ).

% Un_Inter
tff(fact_6997_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X)
             => ? [X2: B] :
                  ( aa(set(B),$o,member(B,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),aa(B,A,F2,X2)) ) ) ) ) ).

% le_SUP_iff
tff(fact_6998_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3))),X)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y5)
             => ? [X2: B] :
                  ( aa(set(B),$o,member(B,X2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_6999_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C2: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,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),image(A,B,F2),I5)) = C2 )
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I5)
                 => ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7000_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),C2: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),C2) )
           => ( ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),I5)) = C2 )
            <=> ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I5)
                 => ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_7001_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),complete_Inf_Inf(A,A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Inf_le_Sup
tff(fact_7002_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_7003_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),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B3))),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_7004_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),$o),ord_less_eq(set(A)),A3),B3)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => 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),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),B3))) ) ) ) ).

% SUP_subset_mono
tff(fact_7005_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),$o),ord_less_eq(set(A)),B3),A3)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),B3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,G),B3))) ) ) ) ).

% INF_superset_mono
tff(fact_7006_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),image(B,A,aTP_Lamp_xp(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(A),C2) ) ).

% SUP_constant
tff(fact_7007_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),image(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_7008_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A3: set(B)] : aa(A,A,uminus_uminus(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B3),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_xq(fun(B,A),fun(B,A),B3)),A3)) ) ).

% uminus_INF
tff(fact_7009_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A3: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B3),A3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_xq(fun(B,A),fun(B,A),B3)),A3)) ) ).

% uminus_SUP
tff(fact_7010_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),X: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xr(B,fun(fun(A,B),fun(A,B)),X),F2)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_7011_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),X: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_xs(fun(A,B),fun(B,fun(A,B)),F2),X)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),I5))),X) ) ) ) ).

% INF_inf_const2
tff(fact_7012_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A2: B,A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),A3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))) ) ).

% SUP_insert
tff(fact_7013_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A2: B,A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),A3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A2)),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),A3))) ) ).

% INF_insert
tff(fact_7014_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M7: fun(B,A),A3: set(B),B3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M7),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M7),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M7),B3))) ) ).

% SUP_union
tff(fact_7015_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M7: fun(B,A),A3: set(B),B3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M7),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M7),A3))),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M7),B3))) ) ).

% INF_union
tff(fact_7016_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(C,set(B)),A3: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),G),A3)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_xt(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F2),G)),A3)) ) ).

% SUP_UNION
tff(fact_7017_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_vz(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3))) ).

% UN_extend_simps(1)
tff(fact_7018_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: 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)),image(B,set(A),A3),C3))),B3) = $ite(C3 = bot_bot(set(B)),B3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3))) ).

% UN_extend_simps(2)
tff(fact_7019_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: 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)),image(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3))) ).

% UN_extend_simps(3)
tff(fact_7020_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3) = $ite(C3 = bot_bot(set(B)),B3,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wz(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3))) ).

% INT_extend_simps(1)
tff(fact_7021_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A3,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wy(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3))) ).

% INT_extend_simps(2)
tff(fact_7022_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C),A7: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( aa(set(A),$o,member(A,I2),I5)
         => ( aa(set(A),$o,member(A,J2),I5)
           => ( aa(set(B),$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),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2)) ) ) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => bij_betw(B,C,F2,aa(A,set(B),A3,I2),aa(A,set(C),A7,I2)) )
       => bij_betw(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),A7),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7023_INT__Un,axiom,
    ! [A: $tType,B: $tType,M7: fun(B,set(A)),A3: set(B),B3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M7),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)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M7),A3))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M7),B3))) ).

% INT_Un
tff(fact_7024_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),minus_minus(set(A),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) ).

% UN_extend_simps(7)
tff(fact_7025_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_7026_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)),complete_Inf_Inf(set(A),A3)),complete_Inf_Inf(set(A),B3))),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_7027_INT__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(set(B))] : complete_Inf_Inf(set(A),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_xv(fun(B,set(A)),fun(set(B),set(A)),B3)),A3)) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% INT_extend_simps(8)
tff(fact_7028_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_xm(fun(B,set(A)),fun(B,set(set(A))),B3)),A3))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3)))) ).

% UN_Pow_subset
tff(fact_7029_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A3: set(A),B14: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),complete_Inf_Inf(set(A),B14)) = $ite(B14 = bot_bot(set(set(A))),A3,complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B14))) ).

% Int_Inter_eq(1)
tff(fact_7030_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B14: set(set(A)),A3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),B14)),A3) = $ite(B14 = bot_bot(set(set(A))),A3,complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_xn(set(A),fun(set(A),set(A)),A3)),B14))) ).

% Int_Inter_eq(2)
tff(fact_7031_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),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% INF_le_SUP
tff(fact_7032_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)),image(B,set(A),aTP_Lamp_xw(fun(B,A),fun(B,set(A)),F2)),A3)) = aa(set(B),set(A),image(B,A,F2),A3) ).

% UNION_singleton_eq_range
tff(fact_7033_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(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)),image(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A3,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3))) ).

% INT_extend_simps(4)
tff(fact_7034_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),image(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_7035_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_xx(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccINF_const
tff(fact_7036_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A2: A,A3: set(A),B3: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B3,A2),B2),C2)
       => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B3),A3)),B2),C2) ) ) ).

% SUP2_I
tff(fact_7037_INF1__I,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(A,fun(B,$o)),B2: B] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(B,$o,aa(A,fun(B,$o),B3,X4),B2) )
     => aa(B,$o,complete_Inf_Inf(fun(B,$o),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B3),A3)),B2) ) ).

% INF1_I
tff(fact_7038_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: set(A),B3: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B3,X4),B2),C2) )
     => aa(C,$o,aa(B,fun(C,$o),complete_Inf_Inf(fun(B,fun(C,$o)),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B3),A3)),B2),C2) ) ).

% INF2_I
tff(fact_7039_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),B3: fun(A,fun(B,$o)),B2: B] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(B,$o,aa(A,fun(B,$o),B3,A2),B2)
       => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B3),A3)),B2) ) ) ).

% SUP1_I
tff(fact_7040_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),image(B,A,aTP_Lamp_xy(B,A)),A3)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_7041_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),image(A,B,aTP_Lamp_xx(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccSUP_const
tff(fact_7042_INF1__D,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B3),A3)),B2)
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(A,$o,aa(B,fun(A,$o),B3,A2),B2) ) ) ).

% INF1_D
tff(fact_7043_INF1__E,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B3),A3)),B2)
     => ( ~ aa(A,$o,aa(B,fun(A,$o),B3,A2),B2)
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INF1_E
tff(fact_7044_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B3),A3)),B2),C2)
     => ( aa(set(C),$o,member(C,A2),A3)
       => aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,A2),B2),C2) ) ) ).

% INF2_D
tff(fact_7045_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B3: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B3),A3)),B2),C2)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,A2),B2),C2)
       => ~ aa(set(C),$o,member(C,A2),A3) ) ) ).

% INF2_E
tff(fact_7046_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,fun(A,$o)),A3: set(B),B2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B3),A3)),B2)
     => ~ ! [X4: B] :
            ( aa(set(B),$o,member(B,X4),A3)
           => ~ aa(A,$o,aa(B,fun(A,$o),B3,X4),B2) ) ) ).

% SUP1_E
tff(fact_7047_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B3),A3)),B2),C2)
     => ~ ! [X4: C] :
            ( aa(set(C),$o,member(C,X4),A3)
           => ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,X4),B2),C2) ) ) ).

% SUP2_E
tff(fact_7048_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% ccpo_Sup_singleton
tff(fact_7049_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),complete_Inf_Inf(A,aa(set(nat),set(A),image(nat,A,aTP_Lamp_xz(A,fun(nat,A),B3)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% INF_nat_binary
tff(fact_7050_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_xz(A,fun(nat,A),B3)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% SUP_nat_binary
tff(fact_7051_Inf__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A2: A,A3: set(A)] : complete_Inf_Inf(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),complete_Inf_Inf(A,A3)) ) ).

% Inf_finite_insert
tff(fact_7052_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A2: A,A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ).

% Sup_finite_insert
tff(fact_7053_UN__UN__split__split__eq,axiom,
    ! [D6: $tType,E4: $tType,A: $tType,C: $tType,B: $tType,A3: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Y4: set(product_prod(D6,E4)),X5: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D6,E4)),fun(B,fun(C,set(A))),aTP_Lamp_ya(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(B,fun(C,set(A)))),A3),Y4))),X5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A)),aTP_Lamp_yd(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A))),A3),Y4)),X5)) ).

% UN_UN_split_split_eq
tff(fact_7054_suminf__eq__SUP__real,axiom,
    ! [X5: fun(nat,real)] :
      ( summable(real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,X5,I2))
       => ( suminf(real,X5) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_ye(fun(nat,real),fun(nat,real),X5)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_7055_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).

% top_apply
tff(fact_7056_UNIV__I,axiom,
    ! [A: $tType,X: A] : aa(set(A),$o,member(A,X),top_top(set(A))) ).

% UNIV_I
tff(fact_7057_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_7058_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_7059_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_7060_Pow__UNIV,axiom,
    ! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_7061_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_yf($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_7062_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_7063_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,$o),ord_less(A),X2),top_top(A))
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_7064_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_7065_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ql(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_7066_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( top_top(A) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B3),A3)) )
        <=> ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A3)
             => ( aa(B,A,B3,X2) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_7067_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B3),A3)) = top_top(A) )
        <=> ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A3)
             => ( aa(B,A,B3,X2) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_7068_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_yg(B,A)),A3)) = top_top(A) ) ).

% INF_top
tff(fact_7069_ccINF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_yh(B,A)),A3)) = top_top(A) ) ).

% ccINF_top
tff(fact_7070_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)) = top_top(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),top_top(A))
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),aa(B,A,F2,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_7071_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_pu(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% range_constant
tff(fact_7072_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_7073_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_vy(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(set(A)),C2) ).

% INT_constant
tff(fact_7074_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),X)
         => ( complete_Inf_Inf(A,aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_7075_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wy(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = 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),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),C3)))) ).

% INT_simps(2)
tff(fact_7076_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: set(A),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wz(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = 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)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3)) ).

% INT_simps(1)
tff(fact_7077_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: set(A),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_xd(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),minus_minus(set(A),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3)) ).

% INT_simps(3)
tff(fact_7078_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(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)),image(B,set(A),B3),C3)))) ).

% INT_simps(4)
tff(fact_7079_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : aa(A,$o,sums(A,F2),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),F2)),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_7080_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_7081_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_7082_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_yk(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = complete_Inf_Inf(A,aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_ym(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_7083_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_yn(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_yo(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_7084_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B3: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))),B3)
     => aa(set(A),$o,member(A,aa(B,A,F2,I)),B3) ) ).

% range_subsetD
tff(fact_7085_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F2,X) )
     => aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_7086_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B] : aa(set(A),$o,member(A,aa(B,A,F2,X)),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_7087_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => ~ ! [X4: B] : B2 != aa(B,A,F2,X4) ) ).

% rangeE
tff(fact_7088_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_yp(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F2),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_7089_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),image(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_yq(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_7090_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(B,A),X5: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image(B,A,F2),X5) = top_top(set(A)) )
     => ( ! [X4: B] :
            ( aa(set(B),$o,member(B,X4),X5)
           => ( aa(B,C,comp(A,C,B,G1,F2),X4) = aa(B,C,comp(A,C,B,G22,F2),X4) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_7091_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( top_top(set(A)) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3)) )
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => ( aa(B,set(A),B3,X2) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_7092_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),A3)) = top_top(set(A)) )
    <=> ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A3)
         => ( aa(B,set(A),B3,X2) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_7093_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_7094_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_7095_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_7096_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_7097_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_7098_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_7099_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_7100_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_7101_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_7102_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_7103_UNIV__eq__I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X4: A] : aa(set(A),$o,member(A,X4),A3)
     => ( top_top(set(A)) = A3 ) ) ).

% UNIV_eq_I
tff(fact_7104_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X4: A] : aa(set(A),$o,member(A,X4),top_top(set(A))) ).

% UNIV_witness
tff(fact_7105_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_rc(A,$o)) ).

% UNIV_def
tff(fact_7106_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_7107_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_7108_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_7109_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_7110_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_7111_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_7112_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_7113_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_7114_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),minus_minus(set(A),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_7115_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_7116_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_7117_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% sup_shunt
tff(fact_7118_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: A,X: B] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) )
     => ( aa(B,A,F2,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_7119_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_7120_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_xp(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(A),C2) ) ).

% INF_constant
tff(fact_7121_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$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),image(B,A,F2),A3))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A3))) ) ).

% surj_Compl_image_subset
tff(fact_7122_INT__empty,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A))] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B3),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_7123_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_yr(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7124_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_7125_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_7126_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B3: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B3) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),B3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_xd(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C3))) ).

% INT_extend_simps(3)
tff(fact_7127_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),B3: fun(C,set(A)),A3: set(C)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image(A,B,F2),complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B3),A3))) = complete_Inf_Inf(set(B),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ys(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B3)),A3)) ) ) ).

% bij_image_INT
tff(fact_7128_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aTP_Lamp_yt(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_7129_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),image(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),image(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_7130_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C3: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C3)
     => aa(set(A),$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)),image(nat,set(A),A3),top_top(set(nat))))),C3) ) ).

% UN_finite_subset
tff(fact_7131_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)),image(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)),image(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)),image(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_7132_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_7133_range__mod,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_yu(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_7134_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),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$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)),image(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_7135_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),F2: fun(A,set(B)),C2: set(B)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A3)
           => ( aa(A,set(B),F2,X4) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),A3)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_7136_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)
       => ( ? [N4: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
                 => ! [M4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N4)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M4)),aa(nat,set(A),F2,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
                 => ( aa(nat,set(A),F2,N4) = 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)),image(nat,set(A),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_7137_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F2,R2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_yv(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R2))) ).

% mlex_eq
tff(fact_7138_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_yw(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_7139_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_yx($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_7140_card__UNIV__bool,axiom,
    aa(set($o),nat,finite_card($o),top_top(set($o))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% card_UNIV_bool
tff(fact_7141_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = zero_zero(real),aa(set(real),set(real),aa(real,fun(set(real),set(real)),insert(real),zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_7142_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : complete_Inf_Inf(A,aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% INF_UNIV_bool_expand
tff(fact_7143_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_7144_Un__eq__UN,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_yy(set(A),fun(set(A),fun($o,set(A))),A3),B3)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_7145_UN__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% UN_bool_eq
tff(fact_7146_INT__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : complete_Inf_Inf(set(A),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% INT_bool_eq
tff(fact_7147_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_7148_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_7149_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_7150_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),top_top(fun(A,$o))) ).

% top_set_def
tff(fact_7151_Id__on__def_H,axiom,
    ! [A: $tType,A3: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A3)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_yz(fun(A,$o),fun(A,fun(A,$o)),A3))) ).

% Id_on_def'
tff(fact_7152_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_7153_int__in__range__abs,axiom,
    ! [Na: nat] : aa(set(int),$o,member(int,aa(nat,int,semiring_1_of_nat(int),Na)),aa(set(int),set(int),image(int,int,abs_abs(int)),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_7154_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R2)) ) ).

% mlex_less
tff(fact_7155_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F2: fun(A,nat),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ).

% mlex_iff
tff(fact_7156_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R2)) ) ) ).

% mlex_leq
tff(fact_7157_Id__on__set,axiom,
    ! [A: $tType,Xsa: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xsa)) = 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_uk(A,product_prod(A,A))),Xsa)) ).

% Id_on_set
tff(fact_7158_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))),image(A,set(product_prod(A,A)),aTP_Lamp_za(A,set(product_prod(A,A)))),A3)) ).

% Id_on_def
tff(fact_7159_root__def,axiom,
    ! [Na: nat,X: real] :
      aa(real,real,root(Na),X) = $ite(Na = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_zb(nat,fun(real,real),Na),X)) ).

% root_def
tff(fact_7160_card__UNIV__char,axiom,
    aa(set(char),nat,finite_card(char),top_top(set(char))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))) ).

% card_UNIV_char
tff(fact_7161_top1I,axiom,
    ! [A: $tType,X: A] : aa(A,$o,top_top(fun(A,$o)),X) ).

% top1I
tff(fact_7162_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Y) ).

% top2I
tff(fact_7163_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_7164_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A3: set(B),F2: fun(B,A),X3: A] : the_inv_into(B,A,A3,F2,X3) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_zc(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A3),F2),X3)) ).

% the_inv_into_def
tff(fact_7165_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7166_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A] : aa(A,char,unique5772411509450598832har_of(A),modulo_modulo(A,Na,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),Na) ) ).

% char_of_mod_256
tff(fact_7167_char__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,char,unique5772411509450598832har_of(nat),Na) ) ).

% char_of_nat
tff(fact_7168_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,Na: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),Na) )
        <=> ( modulo_modulo(A,M,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = modulo_modulo(A,Na,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_7169_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,M: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Na)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7170_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A2)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_7171_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A] : aa(A,char,unique5772411509450598832har_of(A),Na) = aa($o,char,char2(~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),Na),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),one_one(nat)),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2))))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Na),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_7172_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C2),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_7173_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : aa(char,A,comm_s6883823935334413003f_char(A),aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B1)),aa(list($o),list($o),cons($o,(B22)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_7174_char_Osize_I2_J,axiom,
    ! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : aa(char,nat,size_size(char),aa($o,char,char2((X15),(X22),(X32),(X42),(X52),(X62),(X72)),(X8))) = zero_zero(nat) ).

% char.size(2)
tff(fact_7175_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% nat_of_char_less_256
tff(fact_7176_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_7177_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Na) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Na) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_7178_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B22)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B1)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).

% integer_of_char_code
tff(fact_7179_char_Osize__gen,axiom,
    ! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_char(aa($o,char,char2((X15),(X22),(X32),(X42),(X52),(X62),(X72)),(X8))) = zero_zero(nat) ).

% char.size_gen
tff(fact_7180_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_zk(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_7181_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_7182_Gcd__eq__Max,axiom,
    ! [M7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M7)
     => ( ( M7 != bot_bot(set(nat)) )
       => ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M7)
         => ( gcd_Gcd(nat,M7) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),complete_Inf_Inf(set(nat),aa(set(nat),set(set(nat)),image(nat,set(nat),aTP_Lamp_zl(nat,set(nat))),M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7183_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F3: fun(A,filter(B))] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B3)
         => ( aa(set(A),$o,finite_finite2(A),X7)
           => ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),X7)) != bot_bot(filter(B)) ) ) )
     => ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),B3)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_7184_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_7185_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_7186_Gcd__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = gcd_gcd(A,A2,gcd_Gcd(A,A3)) ) ).

% Gcd_insert
tff(fact_7187_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = gcd_gcd(A,A2,B2) ) ).

% Gcd_2
tff(fact_7188_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)),insert(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_7189_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( aa(set(A),$o,member(A,A2),A3)
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_7190_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,member(A,one_one(A)),A3)
         => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_7191_Gcd__nat__eq__one,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,member(nat,one_one(nat)),N3)
     => ( gcd_Gcd(nat,N3) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_7192_Gcd__dvd__nat,axiom,
    ! [A2: nat,A3: set(nat)] :
      ( aa(set(nat),$o,member(nat,A2),A3)
     => dvd_dvd(nat,gcd_Gcd(nat,A3),A2) ) ).

% Gcd_dvd_nat
tff(fact_7193_Gcd__greatest__nat,axiom,
    ! [A3: set(nat),A2: nat] :
      ( ! [B4: nat] :
          ( aa(set(nat),$o,member(nat,B4),A3)
         => dvd_dvd(nat,A2,B4) )
     => dvd_dvd(nat,A2,gcd_Gcd(nat,A3)) ) ).

% Gcd_greatest_nat
tff(fact_7194_Gcd__dvd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,A2),A3)
         => dvd_dvd(A,gcd_Gcd(A,A3),A2) ) ) ).

% Gcd_dvd
tff(fact_7195_dvd__GcdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A3: set(A),Y: A] :
          ( dvd_dvd(A,X,gcd_Gcd(A,A3))
         => ( aa(set(A),$o,member(A,Y),A3)
           => dvd_dvd(A,X,Y) ) ) ) ).

% dvd_GcdD
tff(fact_7196_dvd__Gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A3: set(A)] :
          ( dvd_dvd(A,X,gcd_Gcd(A,A3))
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => dvd_dvd(A,X,X2) ) ) ) ).

% dvd_Gcd_iff
tff(fact_7197_Gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),A2: A] :
          ( ! [B4: A] :
              ( aa(set(A),$o,member(A,B4),A3)
             => dvd_dvd(A,A2,B4) )
         => dvd_dvd(A,A2,gcd_Gcd(A,A3)) ) ) ).

% Gcd_greatest
tff(fact_7198_Gcd__in,axiom,
    ! [A3: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( aa(set(nat),$o,member(nat,A4),A3)
         => ( aa(set(nat),$o,member(nat,B4),A3)
           => aa(set(nat),$o,member(nat,gcd_gcd(nat,A4,B4)),A3) ) )
     => ( ( A3 != bot_bot(set(nat)) )
       => aa(set(nat),$o,member(nat,gcd_Gcd(nat,A3)),A3) ) ) ).

% Gcd_in
tff(fact_7199_Inf__filter__not__bot,axiom,
    ! [A: $tType,B3: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B3)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X7)
           => ( complete_Inf_Inf(filter(A),X7) != bot_bot(filter(A)) ) ) )
     => ( complete_Inf_Inf(filter(A),B3) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_7200_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F3: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),I5)
         => ! [J2: A] :
              ( aa(set(A),$o,member(A,J2),I5)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),I5)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F3,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,I2)),aa(A,filter(B),F3,J2))) ) ) )
     => ( ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),I5)) = bot_bot(filter(B)) )
      <=> ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),I5)
            & ( aa(A,filter(B),F3,X2) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_7201_Gcd__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => dvd_dvd(B,aa(A,B,F2,X4),aa(A,B,G,X4)) )
         => dvd_dvd(B,gcd_Gcd(B,aa(set(A),set(B),image(A,B,F2),A3)),gcd_Gcd(B,aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% Gcd_mono
tff(fact_7202_Gcd__remove0__nat,axiom,
    ! [M7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M7)
     => ( gcd_Gcd(nat,M7) = gcd_Gcd(nat,aa(set(nat),set(nat),minus_minus(set(nat),M7),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_7203_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
         => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_7204_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( X != zero_zero(real) )
       => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
             => ( D = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
               => ( D = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
               => ( D = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Na),D,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_7205_abs__Gcd__eq,axiom,
    ! [K6: set(int)] : aa(int,int,abs_abs(int),gcd_Gcd(int,K6)) = gcd_Gcd(int,K6) ).

% abs_Gcd_eq
tff(fact_7206_Gcd__abs__eq,axiom,
    ! [K6: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image(int,int,abs_abs(int)),K6)) = gcd_Gcd(int,K6) ).

% Gcd_abs_eq
tff(fact_7207_Gcd__nat__abs__eq,axiom,
    ! [K6: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image(int,nat,aTP_Lamp_zm(int,nat)),K6)) = aa(int,nat,nat2,gcd_Gcd(int,K6)) ).

% Gcd_nat_abs_eq
tff(fact_7208_Gcd__int__eq,axiom,
    ! [N3: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image(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_7209_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F3),F9)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F9)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F9),F3) ) ) ).

% less_filter_def
tff(fact_7210_Gcd__int__greater__eq__0,axiom,
    ! [K6: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K6)) ).

% Gcd_int_greater_eq_0
tff(fact_7211_Gcd__dvd__int,axiom,
    ! [A2: int,A3: set(int)] :
      ( aa(set(int),$o,member(int,A2),A3)
     => dvd_dvd(int,gcd_Gcd(int,A3),A2) ) ).

% Gcd_dvd_int
tff(fact_7212_Gcd__greatest__int,axiom,
    ! [A3: set(int),A2: int] :
      ( ! [B4: int] :
          ( aa(set(int),$o,member(int,B4),A3)
         => dvd_dvd(int,A2,B4) )
     => dvd_dvd(int,A2,gcd_Gcd(int,A3)) ) ).

% Gcd_greatest_int
tff(fact_7213_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z2: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z2)),S2)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_zn(fun(A,A),fun(A,fun(A,A)),F2),Z2),Y,topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_at_within_shift
tff(fact_7214_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_zo(fun(A,A),fun(nat,fun(A,A)),F2),Na),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),Na))),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(nat,A,power_power(A,aa(A,A,F2,X)),Na))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power_Suc
tff(fact_7215_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_inverse
tff(fact_7216_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(nat,fun(A,A)),F2),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(nat,A,power_power(A,aa(A,A,F2,X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power
tff(fact_7217_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_mult'
tff(fact_7218_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,X: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,X))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_mult
tff(fact_7219_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_zr(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_cmult
tff(fact_7220_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_zs(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D),C2),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_cmult_right
tff(fact_7221_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_zt(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_7222_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_zu(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_7223_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: filter(A)] : has_field_derivative(A,aTP_Lamp_zv(A,A),one_one(A),F3) ) ).

% DERIV_ident
tff(fact_7224_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F3: filter(A),G: fun(A,A),G3: A] :
          ( has_field_derivative(A,F2,F6,F3)
         => ( has_field_derivative(A,G,G3,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F6),G3),F3) ) ) ) ).

% field_differentiable_add
tff(fact_7225_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D),E5),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_add
tff(fact_7226_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,D),E5),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_diff
tff(fact_7227_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F3: filter(A),G: fun(A,A),G3: A] :
          ( has_field_derivative(A,F2,F6,F3)
         => ( has_field_derivative(A,G,G3,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_zx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,F6),G3),F3) ) ) ) ).

% field_differentiable_diff
tff(fact_7228_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A),X: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),topolo174197925503356063within(A,X,S)),topolo174197925503356063within(A,X,T2)) ) ) ).

% at_le
tff(fact_7229_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S)
           => has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_field_derivative_subset
tff(fact_7230_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,X,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S)
           => has_field_derivative(A,F2,F6,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% DERIV_subset
tff(fact_7231_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,F3: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D,F3)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_zy(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D),F3) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_7232_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_zz(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D),C2),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_cdivide
tff(fact_7233_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_aaa(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_7234_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,F6,F3)
         => has_field_derivative(A,aTP_Lamp_aab(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F6),F3) ) ) ).

% field_differentiable_minus
tff(fact_7235_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_aab(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_minus
tff(fact_7236_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_7237_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_7238_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,minus_minus(real,X),H4)),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H4))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_7239_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,minus_minus(real,X),H4)),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,minus_minus(real,X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_7240_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_aac(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,X))),D)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_7241_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aad(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_7242_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S2: set(A),F2: fun(B,fun(A,B)),F6: fun(C,fun(A,B)),X: C,F3: filter(B)] :
          ( ! [N: A] :
              ( aa(set(A),$o,member(A,N),S2)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_aae(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N),aa(A,B,aa(C,fun(A,B),F6,X),N),F3) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_aaf(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S2),F2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(C,fun(A,B),F6,X)),S2),F3) ) ) ).

% DERIV_sum
tff(fact_7243_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_aag(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,X)))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_7244_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa: A] : has_field_derivative(A,aTP_Lamp_aah(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),K))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_7245_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,minus_minus(real,X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_7246_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H4))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_7247_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                 => 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),X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_7248_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_7249_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_7250_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_field_derivative(real,V2,K,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( aa(real,real,V2,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V2,A2)),aa(real,real,V2,B2))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_7251_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G3: fun(real,real)] :
      ( ! [X4: real] :
          ( aa(set(real),$o,member(real,X4),set_or1337092689740270186AtMost(real,A2,B2))
         => has_field_derivative(real,G,aa(real,real,G3,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ( ! [X4: real] :
            ( aa(set(real),$o,member(real,X4),set_or1337092689740270186AtMost(real,A2,B2))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G3,X4)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G,A2)),aa(real,real,G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_7252_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)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,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_7253_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)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,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_7254_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_7255_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_7256_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
             => has_field_derivative(real,F2,aa(real,real,F6,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
       => ? [Z: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),B2)
            & ( aa(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,minus_minus(real,B2),A2)),aa(real,real,F6,Z)) ) ) ) ) ).

% MVT2
tff(fact_7257_DERIV__ln,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_7258_DERIV__isconst__all,axiom,
    ! [F2: fun(real,real),X: real,Y: real] :
      ( ! [X4: real] : has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real))))
     => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ).

% DERIV_isconst_all
tff(fact_7259_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,X: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),X),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_aai(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_7260_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,Z2: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_aaj(fun(A,A),fun(A,fun(A,A)),F2),Z2),Y,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_7261_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_aak(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_7262_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_aal(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_7263_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S: set(A),G: fun(A,A),G3: fun(A,A),F2: fun(A,A),F6: A,X: A] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),S)
             => has_field_derivative(A,G,aa(A,A,G3,X4),topolo174197925503356063within(A,X4,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( aa(set(A),$o,member(A,aa(A,A,F2,X)),S)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aam(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G3,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_7264_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G3: fun(A,A),F2: fun(A,A),F6: A,X: A] :
          ( ! [X4: A] : has_field_derivative(A,G,aa(A,A,G3,X4),topolo174197925503356063within(A,X4,top_top(set(A))))
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,X,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aam(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G3,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_7265_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aam(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_chain2
tff(fact_7266_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,aa(A,A,F2,X),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aan(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_chain'
tff(fact_7267_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,X: real,D3: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X)),aa(real,real,F2,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_7268_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,X: real,D3: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y3)),aa(real,real,F2,X)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_7269_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,X: real,D3: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D3)
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_7270_DERIV__pow,axiom,
    ! [Na: nat,X: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_aao(nat,fun(real,real),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,X),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S)) ).

% DERIV_pow
tff(fact_7271_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => has_field_derivative(A,aTP_Lamp_aap(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_7272_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B2)
           => ( topolo174197925503356063within(A,X,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,X,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_7273_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,X: real,Na: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_aaq(fun(real,real),fun(nat,fun(real,real)),G),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,G,X)),aa(nat,nat,minus_minus(nat,Na),one_one(nat))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_7274_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_7275_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aad(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,X)))),aa(nat,A,power_power(A,aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_7276_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_aac(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,aa(A,A,F2,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_7277_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F2: fun(A,A),F6: A,Z2: A] :
          ( ! [Z: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),aa(A,A,F2,Z)) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Z2,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_7278_has__real__derivative__powr,axiom,
    ! [Z2: real,R3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z2)
     => has_field_derivative(real,aTP_Lamp_aar(real,fun(real,real),R3),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,Z2,aa(real,real,minus_minus(real,R3),one_one(real)))),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_7279_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aas(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6))
               => has_field_derivative(A,aTP_Lamp_aap(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_7280_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6))
           => has_field_derivative(A,aTP_Lamp_aap(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_7281_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),Z2: A] :
          ( ! [Z: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),Z)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K6)
           => has_field_derivative(A,aTP_Lamp_aap(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),topolo174197925503356063within(A,Z2,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_7282_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,X))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_aat(fun(real,real),fun(real,fun(real,real)),G),R3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,aa(real,real,G,X),aa(real,real,minus_minus(real,R3),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_7283_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => has_field_derivative(real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_7284_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,X))
       => ( has_field_derivative(real,F2,R3,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_aau(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),aa(real,real,ln_ln(real),aa(real,real,G,X)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,X))),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_7285_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_7286_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_7287_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_7288_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_7289_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_7290_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),X: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,X)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aTP_Lamp_aav(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tanh(A),aa(A,A,G,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_7291_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D: real] :
      ( ( X != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( D = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
           => ( D = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_7292_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_7293_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_7294_Gcd__int__def,axiom,
    ! [K6: set(int)] : gcd_Gcd(int,K6) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image(int,nat,comp(int,nat,int,nat2,abs_abs(int))),K6))) ).

% Gcd_int_def
tff(fact_7295_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_7296_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_7297_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_7298_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,Na: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_aaw(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_7299_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,Na: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
          & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_aaw(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_7300_DERIV__odd__real__root,axiom,
    ! [Na: nat,X: real] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_7301_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aax(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_7302_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aax(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_7303_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aax(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_7304_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( X != zero_zero(real) )
         => ( ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
                & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_aaw(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_7305_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,T4: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_aaw(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_7306_Taylor__down,axiom,
    ! [Na: 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)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( 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)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C2)
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aay(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,A2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_7307_Taylor__up,axiom,
    ! [Na: 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)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( 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)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),B2)
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aay(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,B2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_7308_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( 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),X)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
                 => ( ( X != C2 )
                   => ? [T4: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),X) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aaz(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,X),C2)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_7309_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M4: nat,T4: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M2: nat,T8: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T8)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T8),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_abb(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B3),M2),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T8)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_abc(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T8)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,T8),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T8,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_7310_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_abd(real,real),suminf(real,aTP_Lamp_abe(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_7311_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun(nat,real),X0: real] :
      ( ! [X4: real] :
          ( aa(set(real),$o,member(real,X4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_abf(fun(nat,real),fun(real,fun(nat,real)),F2),X4)) )
     => ( aa(set(real),$o,member(real,X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => has_field_derivative(real,aTP_Lamp_abh(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_abf(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_7312_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_abi(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_abj(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_7313_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_7314_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_7315_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_7316_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_7317_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_7318_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_7319_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_7320_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( complete_Inf_Inf(A,set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_7321_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( complete_Inf_Inf(A,set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_7322_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),G: fun(B,C),G3: fun(B,C)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( has_derivative(B,C,G,G3,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_abk(fun(A,B),fun(fun(B,C),fun(A,C)),F6),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_in_compose
tff(fact_7323_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_7324_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abl(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_abm(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),X),G),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_mult
tff(fact_7325_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F6: fun(A,real),X: A,S: set(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abn(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_abo(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),X),G),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_7326_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),T2: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S)
           => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_derivative_subset
tff(fact_7327_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_abp(A,B),F3,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X3: A] : aa(A,B,F3,X3) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_7328_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),G: fun(B,C),G3: fun(B,C)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( has_derivative(B,C,G,G3,topolo174197925503356063within(B,aa(A,B,F2,X),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_abk(fun(A,B),fun(fun(B,C),fun(A,C)),F6),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_compose
tff(fact_7329_has__derivative__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F6,I2),F3) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F6),F3) ) ) ).

% has_derivative_sum
tff(fact_7330_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F6,F3)
         => has_derivative(A,B,aTP_Lamp_abs(fun(A,B),fun(A,B),F2),aTP_Lamp_abs(fun(A,B),fun(A,B),F6),F3) ) ) ).

% has_derivative_minus
tff(fact_7331_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_abt(B,fun(A,B),C2),aTP_Lamp_abp(A,B),F3) ) ).

% has_derivative_const
tff(fact_7332_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : has_derivative(A,A,aTP_Lamp_abu(A,A),aTP_Lamp_abu(A,A),F3) ) ).

% has_derivative_ident
tff(fact_7333_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G3: fun(A,real),F3: filter(A)] :
          ( has_derivative(A,real,G,G3,F3)
         => has_derivative(A,B,aTP_Lamp_abv(fun(A,real),fun(A,B),G),aTP_Lamp_abv(fun(A,real),fun(A,B),G3),F3) ) ) ).

% has_derivative_of_real
tff(fact_7334_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G3: fun(A,B),F3: filter(A),R3: real] :
          ( has_derivative(A,B,G,G3,F3)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_abw(fun(A,B),fun(real,fun(A,B)),G),R3),aa(real,fun(A,B),aTP_Lamp_abw(fun(A,B),fun(real,fun(A,B)),G3),R3),F3) ) ) ).

% has_derivative_scaleR_right
tff(fact_7335_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G3: fun(A,real),F3: filter(A),X: B] :
          ( has_derivative(A,real,G,G3,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_abx(fun(A,real),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_abx(fun(A,real),fun(B,fun(A,B)),G3),X),F3) ) ) ).

% has_derivative_scaleR_left
tff(fact_7336_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F3: filter(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F3)
         => ( has_derivative(A,B,G,G3,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aby(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_aby(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G3),F3) ) ) ) ).

% has_derivative_diff
tff(fact_7337_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F3: filter(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F3)
         => ( has_derivative(A,B,G,G3,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_abz(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G3),F3) ) ) ) ).

% has_derivative_add
tff(fact_7338_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G3: fun(A,B),F3: filter(A),X: B] :
          ( has_derivative(A,B,G,G3,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,B)),G3),X),F3) ) ) ).

% has_derivative_mult_right
tff(fact_7339_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G3: fun(A,B),F3: filter(A),Y: B] :
          ( has_derivative(A,B,G,G3,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_acb(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_acb(fun(A,B),fun(B,fun(A,B)),G3),Y),F3) ) ) ).

% has_derivative_mult_left
tff(fact_7340_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T2: set(A),G: fun(A,B),G3: fun(A,fun(A,B)),F2: fun(C,A),S: set(C),X: C,F6: fun(C,A)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),T2)
             => has_derivative(A,B,G,aa(A,fun(A,B),G3,X4),topolo174197925503356063within(A,X4,T2)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S)),T2)
           => ( aa(set(C),$o,member(C,X),S)
             => ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,X,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acc(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_acd(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G3),F2),X),F6),topolo174197925503356063within(C,X,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_7341_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_ace(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_acf(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G3),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_exp
tff(fact_7342_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_7343_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_acg(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ach(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G3),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_sin
tff(fact_7344_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_zu(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_sinh
tff(fact_7345_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_zt(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_cosh
tff(fact_7346_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_7347_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_7348_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_7349_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_7350_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_7351_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S2: set(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S2))
         => ( has_derivative(A,B,G,G3,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aci(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_acj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),X),G),G3),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_7352_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S2: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_ack(A,fun(A,A),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_inverse'
tff(fact_7353_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),X: B,F6: fun(B,A),S2: set(B)] :
          ( ( aa(B,A,F2,X) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F6,topolo174197925503356063within(B,X,S2))
           => has_derivative(B,A,aTP_Lamp_acl(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_acm(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),X),F6),topolo174197925503356063within(B,X,S2)) ) ) ) ).

% has_derivative_inverse
tff(fact_7354_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F6: real,G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( has_field_derivative(real,F2,F6,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))))
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_acn(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_aco(real,fun(fun(A,real),fun(A,real)),F6),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_7355_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_acp(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_acq(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G3),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_cos
tff(fact_7356_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(set(real),$o,member(real,X),set_or5935395276787703475ssThan(real,A2,B2))
       => ( aa(set(real),$o,member(real,Y),set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X4: real] :
                ( aa(set(real),$o,member(real,X4),set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) )
           => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_7357_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_7358_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_7359_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S2: set(A),Na: nat] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_acr(fun(A,B),fun(nat,fun(A,B)),F2),Na),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_acs(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F6),X),Na),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_power
tff(fact_7360_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_act(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_acu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_ln
tff(fact_7361_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S2: set(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S2))
         => ( has_derivative(A,B,G,G3,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acv(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_acw(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),X),G),G3),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_7362_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),X: B,S2: set(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F6,I2),topolo174197925503356063within(B,X,S2)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_acy(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_ada(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F6),X),topolo174197925503356063within(B,X,S2)) ) ) ).

% has_derivative_prod
tff(fact_7363_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,X5: set(A),F2: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,X5))
         => ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,X,X5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X))
             => ( aa(set(A),$o,member(A,X),X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adb(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_adc(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G3),X),F2),F6),topolo174197925503356063within(A,X,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_7364_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_add(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_ade(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_7365_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_adf(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_adg(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G3),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_arctan
tff(fact_7366_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_adh(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_adi(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_tan
tff(fact_7367_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F6: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_adj(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N),aa(nat,real,aa(real,fun(nat,real),F6,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X4: real] :
            ( aa(set(real),$o,member(real,X4),set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F2,X4)) )
       => ( aa(set(real),$o,member(real,X0),set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F6,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X4: real,Y3: real] :
                    ( aa(set(real),$o,member(real,X4),set_or5935395276787703475ssThan(real,A2,B2))
                   => ( aa(set(real),$o,member(real,Y3),set_or5935395276787703475ssThan(real,A2,B2))
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),F2,X4),N)),aa(nat,real,aa(real,fun(nat,real),F2,Y3),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N)),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X4),Y3)))) ) )
               => has_field_derivative(real,aTP_Lamp_adk(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F6,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_7368_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_adl(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_adm(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_7369_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),X: B,F2: fun(real,A),G3: fun(B,real),S: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,X),top_top(set(real))),F2)
         => ( ~ aa(set(A),$o,member(A,aa(real,A,F2,aa(B,real,G,X))),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G3,topolo174197925503356063within(B,X,S))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_adn(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_ado(fun(B,real),fun(B,real),G3),topolo174197925503356063within(B,X,S)) ) ) ) ) ).

% has_derivative_floor
tff(fact_7370_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aas(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_adq(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_7371_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F3: filter(A)] : filterlim(A,B,aTP_Lamp_adr(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F3) ) ).

% tendsto_const
tff(fact_7372_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S: set(A)] : filterlim(A,A,aTP_Lamp_ads(A,A),topolo7230453075368039082e_nhds(A,A2),topolo174197925503356063within(A,A2,S)) ) ).

% tendsto_ident_at
tff(fact_7373_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_adt(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_7374_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_adu(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_7375_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(nat,fun(fun(A,real),fun(A,real)),Na),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_7376_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [X: A,S: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,S),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,X,S),aa(fun(B,C),fun(A,C),aTP_Lamp_adw(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_7377_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),X: A,S2: set(A),T5: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,X,S2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S2)
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,X,T5)) ) ) ) ).

% tendsto_within_subset
tff(fact_7378_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S2: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),aTP_Lamp_adx(A,A)) ) ).

% continuous_ident
tff(fact_7379_filterlim__at__within__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G4: filter(B),X: A,A3: set(A),P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G4,topolo174197925503356063within(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))))
         => ( filterlim(A,B,G,G4,topolo174197925503356063within(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ady(fun(A,$o),fun(A,$o),P)))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_adz(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G4,topolo174197925503356063within(A,X,A3)) ) ) ) ).

% filterlim_at_within_If
tff(fact_7380_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S2))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aea(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_field_derivativeD
tff(fact_7381_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S2))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aea(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_field_derivative_iff
tff(fact_7382_isCont__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aeb(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% isCont_snd
tff(fact_7383_isCont__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aec(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% isCont_fst
tff(fact_7384_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aed(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aee(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_7385_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aef(A,fun(fun(A,B),fun(A,B)),X),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_7386_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_aeg(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_7387_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_aeg(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_7388_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_aeg(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_7389_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),D2) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aeh(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_7390_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),D2) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aeh(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_7391_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R3: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X3: A] :
                    ( ( ( X3 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L5))),R3) ) ) ) ) ) ).

% LIM_D
tff(fact_7392_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X4)),L5))),R) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_7393_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A2))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X2)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_7394_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R2: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),R2)
                 => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_7395_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X4)) )
           => ( ! [X4: A] :
                  ( ( X4 != A2 )
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,G,X4)),aa(A,real,F2,X4)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_7396_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_aei(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_7397_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aej(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_7398_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G4: filter(B),X: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G4,topolo174197925503356063within(A,X,aa(fun(A,$o),set(A),collect(A),P)))
         => ( filterlim(A,B,G,G4,topolo174197925503356063within(A,X,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ady(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_adz(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G4,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_7399_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_adw(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_7400_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,A2: A] :
          ( filterlim(A,B,aTP_Lamp_aek(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_7401_tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(A,B),L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% tendsto_compose
tff(fact_7402_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,A2: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_aem(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_7403_isCont__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [L: A,G: fun(A,B),F2: fun(C,A),F3: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aen(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% isCont_tendsto_compose
tff(fact_7404_continuous__within__compose3,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topological_t2_space(C) )
     => ! [F2: fun(C,A),X: C,G: fun(A,B),S: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,X),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,S),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,X,S),aa(fun(A,B),fun(C,B),aTP_Lamp_aeo(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_7405_isCont__norm,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)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aep(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_7406_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,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_aeq(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_7407_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,$o),ord_less_eq(B),A2),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F2) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_7408_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,$o),ord_less_eq(B),A2),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F2) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_7409_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X4: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
       => ? [L6: real,M8: 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) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(real,real,F2,X3))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X3)),M8) ) )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),M8) )
               => ? [X4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                    & ( aa(real,real,F2,X4) = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_7410_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,minus_minus(C,aa(A,C,G,X4)),M))),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X4)),L))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_7411_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_aer(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_7412_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)
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X3))),R) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_7413_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) )
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X3))),R) )
               => ( aa(real,real,F2,X3) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_7414_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))
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X3))),R) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X3)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_7415_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aes(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_7416_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$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_aet(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_7417_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( aa(real,$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_aeu(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_7418_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_aev(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ).

% tendsto_arcosh
tff(fact_7419_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_aew(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_7420_has__derivative__Re,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G3: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G3,F3)
         => has_derivative(A,real,aTP_Lamp_aex(fun(A,complex),fun(A,real),G),aTP_Lamp_aex(fun(A,complex),fun(A,real),G3),F3) ) ) ).

% has_derivative_Re
tff(fact_7421_has__derivative__Im,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G3: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G3,F3)
         => has_derivative(A,real,aTP_Lamp_aey(fun(A,complex),fun(A,real),G),aTP_Lamp_aey(fun(A,complex),fun(A,real),G3),F3) ) ) ).

% has_derivative_Im
tff(fact_7422_has__derivative__cnj,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G3: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G3,F3)
         => has_derivative(A,complex,aTP_Lamp_aez(fun(A,complex),fun(A,complex),G),aTP_Lamp_aez(fun(A,complex),fun(A,complex),G3),F3) ) ) ).

% has_derivative_cnj
tff(fact_7423_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: A,A3: set(A),F2: fun(B,C),F3: filter(C),G4: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,X),A3)
     => ( filterlim(B,C,F2,F3,aa(A,filter(B),G4,X))
       => filterlim(B,C,F2,F3,complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),G4),A3))) ) ) ).

% filterlim_INF'
tff(fact_7424_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G4: fun(C,filter(B)),B3: set(C),F3: filter(A)] :
      ( filterlim(A,B,F2,complete_Inf_Inf(filter(B),aa(set(C),set(filter(B)),image(C,filter(B),G4),B3)),F3)
    <=> ! [X2: C] :
          ( aa(set(C),$o,member(C,X2),B3)
         => filterlim(A,B,F2,aa(C,filter(B),G4,X2),F3) ) ) ).

% filterlim_INF
tff(fact_7425_isCont__real__sqrt,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_7426_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F33: filter(B),F23: filter(A),F2: fun(C,A),F12: filter(C)] :
      ( filterlim(A,B,G,F33,F23)
     => ( filterlim(C,A,F2,F23,F12)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afa(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F33,F12) ) ) ).

% filterlim_compose
tff(fact_7427_filterlim__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filterlim(A,A,aTP_Lamp_ab(A,A),F3,F3) ).

% filterlim_ident
tff(fact_7428_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F3: filter(A),A2: B,B2: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_afb(B,fun(A,B),A2),topolo7230453075368039082e_nhds(B,B2),F3)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_7429_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_7430_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S2: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),S2)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aff(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S2),F2)) ) ) ).

% continuous_prod
tff(fact_7431_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afh(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,A2,I5)),F3) ) ) ).

% tendsto_prod'
tff(fact_7432_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S2: set(A),F2: fun(A,fun(B,C)),L5: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),S2)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afj(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S2),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L5,S2)),F3) ) ) ).

% tendsto_prod
tff(fact_7433_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_afk(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_afl(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) ) ) ).

% tendsto_one_prod'
tff(fact_7434_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_afm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_7435_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),X: B,Net: filter(A),Y4: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,X),Net)
         => ( filterlim(A,B,Y4,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afn(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y4),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_7436_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,C,F3,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F3,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aes(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_7437_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F3)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_afo(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),F3) ) ) ) ).

% tendsto_Pair
tff(fact_7438_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_afp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_7439_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_afp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_7440_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_afp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_7441_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F3)
     => filterlim(A,real,aTP_Lamp_afp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F3) ) ).

% tendsto_rabs
tff(fact_7442_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_7443_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_afr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_7444_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_afs(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_7445_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_aft(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_7446_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_afu(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_7447_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_afv(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_7448_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_afw(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_7449_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),F3) ) ) ) ).

% tendsto_mult
tff(fact_7450_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_afy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_7451_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D3: B,F3: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afz(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),D3)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D3),F3) ) ) ).

% tendsto_add_const_iff
tff(fact_7452_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aga(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)),F3) ) ) ) ).

% tendsto_add
tff(fact_7453_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_agb(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_7454_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,minus_minus(B,A2),B2)),F3) ) ) ) ).

% tendsto_diff
tff(fact_7455_lim__cnj,axiom,
    ! [A: $tType,F2: fun(A,complex),L: complex,F3: filter(A)] :
      ( filterlim(A,complex,aTP_Lamp_ni(fun(A,complex),fun(A,complex),F2),topolo7230453075368039082e_nhds(complex,cnj(L)),F3)
    <=> filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,L),F3) ) ).

% lim_cnj
tff(fact_7456_tendsto__cnj,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F3)
     => filterlim(A,complex,aTP_Lamp_ni(fun(A,complex),fun(A,complex),G),topolo7230453075368039082e_nhds(complex,cnj(A2)),F3) ) ).

% tendsto_cnj
tff(fact_7457_tendsto__complex__iff,axiom,
    ! [A: $tType,F2: fun(A,complex),X: complex,F3: filter(A)] :
      ( filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,X),F3)
    <=> ( filterlim(A,real,aTP_Lamp_ms(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,re(X)),F3)
        & filterlim(A,real,aTP_Lamp_nf(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,im(X)),F3) ) ) ).

% tendsto_complex_iff
tff(fact_7458_tendsto__Im,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F3)
     => filterlim(A,real,aTP_Lamp_nf(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,im(A2)),F3) ) ).

% tendsto_Im
tff(fact_7459_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A),Na: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_agd(fun(A,real),fun(nat,fun(A,real)),F2),Na),topolo7230453075368039082e_nhds(real,aa(real,real,root(Na),X)),F3) ) ).

% tendsto_real_root
tff(fact_7460_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_age(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sinh
tff(fact_7461_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_agf(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sinh(B,A2)),F3) ) ) ).

% tendsto_sinh
tff(fact_7462_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_agg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cosh(B,A2)),F3) ) ) ).

% tendsto_cosh
tff(fact_7463_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agh(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,A2),B2)),F3) ) ) ) ).

% tendsto_scaleR
tff(fact_7464_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_aeq(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_7465_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B2),F3)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_agi(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A2),B2)),F3) ) ) ) ).

% tendsto_power_strong
tff(fact_7466_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,nat,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,nat),fun(A,B),aTP_Lamp_agj(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_7467_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_agk(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A2),Na)),F3) ) ) ).

% tendsto_power
tff(fact_7468_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(nat,fun(A,B),aTP_Lamp_agl(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_power
tff(fact_7469_tendsto__arsinh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => filterlim(A,real,aTP_Lamp_agm(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A2)),F3) ) ).

% tendsto_arsinh
tff(fact_7470_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_agn(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sin(B,A2)),F3) ) ) ).

% tendsto_sin
tff(fact_7471_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_ago(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cos(B,A2)),F3) ) ) ).

% tendsto_cos
tff(fact_7472_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aep(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_7473_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A2)),F3) ) ) ).

% tendsto_norm
tff(fact_7474_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_agq(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(A2,B2)),F3) ) ) ).

% tendsto_Complex
tff(fact_7475_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_agr(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_7476_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,real),C2: real,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_ags(fun(A,real),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F3)
        <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3) ) ) ).

% tendsto_of_real_iff
tff(fact_7477_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,G)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aej(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_7478_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),A2: real,F3: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,A2),F3)
         => filterlim(A,B,aTP_Lamp_agt(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),A2)),F3) ) ) ).

% tendsto_of_real
tff(fact_7479_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_agu(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,X)),F3) ) ).

% tendsto_arctan
tff(fact_7480_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_agv(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,X)),F3) ) ).

% tendsto_real_sqrt
tff(fact_7481_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_agw(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),A2)),F3) ) ) ).

% tendsto_exp
tff(fact_7482_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_agx(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_exp
tff(fact_7483_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_agn(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_7484_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_ago(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_7485_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_agy(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cosh
tff(fact_7486_tendsto__Re,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F3)
     => filterlim(A,real,aTP_Lamp_ms(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,re(A2)),F3) ) ).

% tendsto_Re
tff(fact_7487_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_agz(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F3) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_7488_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_aha(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F3) ) ) ) ).

% tendsto_of_int_floor
tff(fact_7489_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahb(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_7490_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ahc(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_7491_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ahd(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_7492_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ahe(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_7493_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr
tff(fact_7494_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( ( A2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_ly(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A2)),F3) ) ) ).

% tendsto_ln
tff(fact_7495_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_7496_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_7497_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_7498_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_ahg(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F3) ) ) ) ).

% tendsto_tan
tff(fact_7499_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ahh(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_7500_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_ahi(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F3) ) ) ) ).

% tendsto_cot
tff(fact_7501_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_ahj(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F3) ) ) ).

% tendsto_minus
tff(fact_7502_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_ahj(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ).

% tendsto_minus_cancel
tff(fact_7503_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F3)
        <=> filterlim(A,B,aTP_Lamp_ahj(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F3) ) ) ).

% tendsto_minus_cancel_left
tff(fact_7504_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_ahk(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_7505_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_7506_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ahm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_7507_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform2
tff(fact_7508_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F3: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahn(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform
tff(fact_7509_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ahm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_7510_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ahm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_7511_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aga(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_7512_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_aho(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_7513_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ahp(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_7514_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_7515_tendsto__fst,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),A2: product_prod(B,C),F3: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),A2),F3)
         => filterlim(A,B,aTP_Lamp_ahr(fun(A,product_prod(B,C)),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(product_prod(B,C),B,product_fst(B,C),A2)),F3) ) ) ).

% tendsto_fst
tff(fact_7516_tendsto__snd,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),A2: product_prod(B,C),F3: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),A2),F3)
         => filterlim(A,C,aTP_Lamp_ahs(fun(A,product_prod(B,C)),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(product_prod(B,C),C,product_snd(B,C),A2)),F3) ) ) ).

% tendsto_snd
tff(fact_7517_continuous__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aec(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% continuous_fst
tff(fact_7518_continuous__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F3,F2)
         => topolo3448309680560233919inuous(A,C,F3,aTP_Lamp_aeb(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% continuous_snd
tff(fact_7519_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_aht(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ahu(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_7520_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ahw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_7521_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ahy(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),A2),I5)),F3) ) ) ).

% tendsto_sum
tff(fact_7522_tendsto__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: filter(A),F9: filter(A),F2: fun(A,B),L: B] :
          ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F9)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F9)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% tendsto_mono
tff(fact_7523_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A),F24: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F23),F24)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F13),F12)
         => filterlim(A,B,F2,F24,F13) ) ) ) ).

% filterlim_mono
tff(fact_7524_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F33: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F33),F12)
    <=> ( filterlim(A,B,F2,F23,F12)
        & filterlim(A,B,F2,F33,F12) ) ) ).

% filterlim_inf
tff(fact_7525_isCont__real__root,axiom,
    ! [X: real,Na: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),root(Na)) ).

% isCont_real_root
tff(fact_7526_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_ahz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_7527_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_afr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_7528_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_afy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_7529_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aia(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_7530_isCont__minus,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)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aib(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_7531_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_agl(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% isCont_power
tff(fact_7532_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_aic(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_7533_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A3: set(A),A2: B,F2: fun(A,fun(B,C))] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(A,fun(B,C),F2,X4)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ahw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A3),F2)) ) ) ).

% isCont_sum
tff(fact_7534_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_aid(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_7535_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_ago(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_7536_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_agn(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_7537_isCont__exp_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_agx(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_exp'
tff(fact_7538_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z2: A,Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z2,top_top(set(A))),aTP_Lamp_aie(nat,fun(A,A),Na)) ) ).

% isCont_pochhammer
tff(fact_7539_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aif(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_7540_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aif(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_7541_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_aig(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_7542_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : set_or5935395276787703475ssThan(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_7543_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K6),real_V7770717601297561774m_norm(A,H3))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_7544_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),M8),aa(real,A,F2,X3)) )
                & ? [X4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                    & ( aa(real,A,F2,X4) = M8 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_7545_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),aa(real,A,F2,X3)),M8) )
                & ? [X4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                    & ( aa(real,A,F2,X4) = M8 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_7546_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
              ! [X3: real] :
                ( ( aa(real,$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_eq(A),aa(real,A,F2,X3)),M8) ) ) ) ) ).

% isCont_bounded
tff(fact_7547_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),B2)
       => ( ! [Z: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z),B2)
               => ( aa(real,real,G,aa(real,real,F2,Z)) = Z ) ) )
         => ( ! [Z: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z),B2)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_7548_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aif(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_7549_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_7550_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_ahz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_7551_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_aid(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_7552_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aih(fun(A,B),fun(A,fun(A,B)),F2),A2),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_7553_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_ahg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_7554_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_ahi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_7555_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [X: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,A3),F2)
         => ( ( cosh(B,aa(A,B,F2,X)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,A3),aTP_Lamp_aii(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_7556_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),aa(real,A,F2,X3)),M8) )
                & ! [N4: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),M8)
                   => ? [X4: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),aa(real,A,F2,X4)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_7557_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_7558_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_7559_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_7560_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7561_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,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),A2),X4)),aa(A,A,F2,X4)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_7562_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,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X4: A] :
                ( ( X4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),A2),X4)),aa(A,A,F2,X4)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_7563_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,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H3: A,N: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H3),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H3))) ) )
             => filterlim(A,B,aTP_Lamp_aij(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_7564_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_ahg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_7565_isCont__arcosh,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_7566_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_aik(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_7567_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_ahi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_7568_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D: real,G: fun(real,real),X: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),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,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_7569_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_ail(fun(nat,A),fun(nat,fun(A,A)),C2),Na)) ) ).

% isCont_polynom
tff(fact_7570_isCont__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_7571_isCont__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_7572_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_aap(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_7573_LIM__less__bound,axiom,
    ! [B2: real,X: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),X)
     => ( ! [X4: real] :
            ( aa(set(real),$o,member(real,X4),set_or5935395276787703475ssThan(real,B2,X))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X4)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X)) ) ) ) ).

% LIM_less_bound
tff(fact_7574_isCont__artanh,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_7575_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_7576_isCont__inverse__function,axiom,
    ! [D3: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
     => ( ! [Z: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z),X))),D3)
           => ( aa(real,real,G,aa(real,real,F2,Z)) = Z ) )
       => ( ! [Z: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z),X))),D3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_7577_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G3: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [Z: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z),B2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),F2) ) )
       => ( ! [Z: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),G) ) )
         => ( ! [Z: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),B2)
                 => has_field_derivative(real,G,aa(real,real,G3,Z),topolo174197925503356063within(real,Z,top_top(set(real)))) ) )
           => ( ! [Z: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),B2)
                   => has_field_derivative(real,F2,aa(real,real,F6,Z),topolo174197925503356063within(real,Z,top_top(set(real)))) ) )
             => ? [C5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C5),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G3,C5)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F6,C5)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_7578_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7579_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [X: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => ( ~ aa(set(A),$o,member(A,aa(real,A,F2,X)),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_aim(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_7580_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),C2: fun(nat,B),K6: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_ain(fun(nat,B),fun(B,fun(nat,B)),C2),K6))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,A2))),real_V7770717601297561774m_norm(B,K6))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_aip(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_7581_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_aap(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_7582_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))
         => ! [N8: nat] : aa(set(real),$o,member(real,suminf(real,aTP_Lamp_aiq(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_aiq(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))),N8)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(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))),N8))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7583_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)))
         => ! [N8: nat] : aa(set(real),$o,member(real,suminf(real,aTP_Lamp_aiq(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_aiq(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))),N8))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(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))),N8)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7584_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_air(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_7585_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_ais(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_7586_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_ait(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_7587_isCont__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aiu(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Re
tff(fact_7588_isCont__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aiv(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Im
tff(fact_7589_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ? [U2: fun(nat,A)] :
              ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,U2,N8))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_7590_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ? [U2: fun(nat,A)] :
              ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U2,N8)),X)
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_7591_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_rq(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7592_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aiw(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_7593_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_aix(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_7594_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_aix(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_7595_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aiy(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_7596_continuous__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,complex,F3,aTP_Lamp_aiz(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_cnj
tff(fact_7597_continuous__complex__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,F2)
        <=> ( topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aiu(fun(A,complex),fun(A,real),F2))
            & topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aiv(fun(A,complex),fun(A,real),F2)) ) ) ) ).

% continuous_complex_iff
tff(fact_7598_continuous__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aiv(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Im
tff(fact_7599_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aja(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_7600_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),Na: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_ajb(fun(A,real),fun(nat,fun(A,real)),F2),Na)) ) ) ).

% continuous_real_root
tff(fact_7601_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajc(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_7602_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_ajd(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_7603_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_ajd(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_7604_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_aje(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_7605_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajf(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_7606_continuous__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aiu(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Re
tff(fact_7607_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X5: fun(nat,A),Y4: fun(nat,A),X: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y4,N)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y4,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ).

% lim_mono
tff(fact_7608_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,Y4: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y4,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N4: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y4,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_7609_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M7: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),C3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C3) ) ) ) ).

% Lim_bounded
tff(fact_7610_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N3: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(nat,A,F2,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),L) ) ) ) ).

% Lim_bounded2
tff(fact_7611_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N4: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
               => aa(A,$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),X) ) ) ) ).

% LIMSEQ_le_const
tff(fact_7612_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N4: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
               => aa(A,$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),X),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_7613_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N: nat] : aa(set(A),$o,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_7614_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N: nat] : aa(set(A),$o,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),complete_Inf_Inf(A,S)),A2) ) ) ) ).

% Inf_lim
tff(fact_7615_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_7616_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aiw(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_7617_isCont__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aiz(fun(A,complex),fun(A,complex),G)) ) ) ).

% isCont_cnj
tff(fact_7618_continuous__at__within__powr,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(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_ajg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_7619_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),aTP_Lamp_ajh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_7620_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_aji(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_7621_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_7622_monoseq__convergent,axiom,
    ! [X5: fun(nat,real),B3: real] :
      ( topological_monoseq(real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X5,I2))),B3)
       => ~ ! [L6: real] : ~ filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_7623_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),X: A] :
          ( filterlim(nat,A,aTP_Lamp_ajj(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_ajk(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_7624_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_ajl(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_7625_isCont__powr,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(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_ajg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_7626_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_ajh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_7627_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),X: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N8)),X)
                & ! [M2: nat,N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N8)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N8)) ) )
              | ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(nat,A,A2,N8))
                & ! [M2: nat,N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N8)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N8)),aa(nat,A,A2,M2)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_7628_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_ajm(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_7629_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),L: B] :
          ( ! [S4: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(nat,A,S4,N8) != A2
                & filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ajn(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S4),topolo7230453075368039082e_nhds(B,L),at_top(nat)) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_7630_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ! [S9: fun(nat,A)] :
              ( ( ! [N: nat] : aa(nat,A,S9,N) != A2
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ajo(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S9),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_7631_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,X5: fun(A,B),L5: B] :
          ( ! [S10: fun(nat,A)] :
              ( ( ! [N2: nat] : aa(nat,A,S10,N2) != A2
                & filterlim(nat,A,S10,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ajn(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S10),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_7632_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ajp(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_7633_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X5: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ajq(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_7634_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_ajr(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_7635_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_ajs(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_7636_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,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,G,aa(nat,nat,suc,N))),aa(nat,real,G,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,G,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ajt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N8)),L2)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(nat,real,G,N8))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7637_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R: real] :
        ? [N4: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),aa(nat,real,X5,N)) )
     => filterlim(nat,real,aTP_Lamp_aju(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_7638_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_ajv(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_7639_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_ajw(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_7640_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_ajx(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7641_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_ajy(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7642_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_ajz(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def
tff(fact_7643_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_aka(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_7644_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_akb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_7645_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)
             => ? [N8: 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,N8)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7646_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_akc(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_7647_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_akd(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_7648_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ake(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_7649_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_ajr(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_7650_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_ajs(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_7651_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => filterlim(nat,real,power_power(real,X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_7652_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_akf(real,fun(real,fun(nat,real)),X),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_7653_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_7654_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_7655_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => filterlim(nat,real,aTP_Lamp_akg(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_7656_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_akh(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_7657_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_aki(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7658_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_akj(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_7659_summable__LIMSEQ,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_akk(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_7660_summable__LIMSEQ_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_akl(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_7661_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_akb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_7662_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => ? [No: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N8)),L5))),R3) ) ) ) ) ).

% LIMSEQ_D
tff(fact_7663_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$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,minus_minus(A,aa(nat,A,X5,N)),L5))),R) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_7664_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,minus_minus(A,aa(nat,A,X5,N2)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_7665_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real))
         => filterlim(nat,A,power_power(A,X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_7666_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_akm(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_7667_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: B,X: A,S: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,X,S))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I4: nat] : aa(set(A),$o,member(A,aa(nat,A,X9,I4)),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,X),at_top(nat))
               => filterlim(nat,B,comp(A,B,nat,F2,X9),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_7668_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F3: filter(A),X: B] :
          ( filterlim(A,nat,F2,at_top(nat),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,X)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_akn(fun(A,nat),fun(B,fun(A,B)),F2),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_7669_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_ako(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7670_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))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_7671_summable__Leibniz_I1_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => summable(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_7672_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z2: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z2,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_akp(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z2),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_7673_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_akq(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_7674_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X))
         => filterlim(nat,A,aTP_Lamp_akr(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_7675_summable,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))
         => summable(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_7676_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_aks(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_akt(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_7677_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_aku(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_akt(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_7678_summable__Leibniz_I4_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_akv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_7679_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_ao(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_7680_summable__Leibniz_H_I3_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_akv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7681_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Na: 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),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(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))),Na)))),suminf(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7682_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))
         => ? [L2: real] :
              ( ! [N8: 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_aiq(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))),N8)))),L2)
              & filterlim(nat,real,aTP_Lamp_akv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(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))),N8)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_akw(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7683_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_akw(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_7684_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Na: 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_aiq(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_aiq(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))),Na)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_7685_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_akw(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7686_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_akx(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_7687_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_aky(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_7688_bounded__linear_Ohas__derivative,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),G3: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G3,F3)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_acc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G3),F3) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_7689_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F3,G)
           => topolo3448309680560233919inuous(C,B,F3,aa(fun(C,A),fun(C,B),aTP_Lamp_akz(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_7690_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),A2: A,F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A2),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ala(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),F3) ) ) ) ).

% bounded_linear.tendsto
tff(fact_7691_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
            ! [X3: A] : aa(real,$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)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_7692_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_alb(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_7693_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),X: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_7694_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_acb(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_7695_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),X)) ) ).

% bounded_linear_mult_right
tff(fact_7696_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_abz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_7697_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aby(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_7698_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => ( aa(A,B,F2,suminf(A,X5)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alc(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_7699_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A),A2: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( aa(A,$o,sums(A,X5),A2)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alc(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)),aa(A,B,F2,A2)) ) ) ) ).

% bounded_linear.sums
tff(fact_7700_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alc(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_7701_bounded__linear__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => real_V3181309239436604168linear(real,A,real_Vector_of_real(A)) ) ).

% bounded_linear_of_real
tff(fact_7702_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alc(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.summable
tff(fact_7703_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_ald(A,fun(real,A),X)) ) ).

% bounded_linear_scaleR_left
tff(fact_7704_bounded__linear__const__scaleR,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),R3: real] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(real,fun(A,B),aTP_Lamp_abw(fun(A,B),fun(real,fun(A,B)),G),R3)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_7705_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R3)) ) ).

% bounded_linear_scaleR_right
tff(fact_7706_bounded__linear__scaleR__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,real),X: B] :
          ( real_V3181309239436604168linear(A,real,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_abx(fun(A,real),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_7707_bounded__linear__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(C,A,G)
           => real_V3181309239436604168linear(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_7708_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_abu(A,A)) ) ).

% bounded_linear_ident
tff(fact_7709_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ale(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_7710_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_abp(A,B)) ) ).

% bounded_linear_zero
tff(fact_7711_bounded__linear__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => real_V3181309239436604168linear(A,B,aTP_Lamp_abs(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_7712_bounded__linear__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),F2,I2)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_7713_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ala(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_7714_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),A2: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,A2,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_akz(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_7715_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [X3: A] : aa(real,$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)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_7716_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,X,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_7717_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,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X3: A] : aa(real,$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)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_7718_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K6: real] :
          ( ! [X4: A,Y3: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3))
         => ( ! [R: real,X4: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R),X4)) = aa(B,B,real_V8093663219630862766scaleR(B,R),aa(A,B,F2,X4))
           => ( ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K6))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_7719_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_alf(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F6),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_7720_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_alg(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_7721_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F6: fun(A,B),X: A,F2: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F6)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_alh(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F6),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivativeI
tff(fact_7722_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E3: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),aa(A,B,F6,H5))),aa(A,B,E3,H5))
                & filterlim(A,real,aTP_Lamp_ali(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_7723_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_akx(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_within
tff(fact_7724_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F6,F3)
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_alj(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F6),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_7725_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S2: set(A),F2: fun(A,B),F6: fun(A,B)] :
          ( aa(set(A),$o,member(A,X),S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S2))
            <=> ( real_V3181309239436604168linear(A,B,F6)
                & ? [E3: fun(A,B)] :
                    ( ! [H5: A] :
                        ( aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)),S2)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),aa(A,B,F6,H5))),aa(A,B,E3,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_ali(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_7726_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_aje(A,fun(nat,A),A2)) = A2 ) ).

% lim_const
tff(fact_7727_open__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B3: fun(A,set(B))] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A3)
             => topolo1002775350975398744n_open(B,aa(A,set(B),B3,X4)) )
         => topolo1002775350975398744n_open(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ) ).

% open_UN
tff(fact_7728_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)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => topolo1002775350975398744n_open(B,aa(A,set(B),B3,X4)) )
           => topolo1002775350975398744n_open(B,complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B3),A3))) ) ) ) ).

% open_INT
tff(fact_7729_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
        ? [A8: fun(nat,set(A))] :
          ( ! [I3: nat] :
              ( aa(set(A),$o,member(A,X),aa(nat,set(A),A8,I3))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A8,I3)) )
          & ! [S9: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S9)
                & aa(set(A),$o,member(A,X),S9) )
             => ? [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),A8,I2)),S9) ) ) ) ).

% first_countable_basis
tff(fact_7730_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),S2)
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & aa(set(A),$o,member(A,X2),T9)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),S2) ) ) ) ) ).

% open_subopen
tff(fact_7731_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),S2)
             => ? [T10: set(A)] :
                  ( topolo1002775350975398744n_open(A,T10)
                  & aa(set(A),$o,member(A,X4),T10)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T10),S2) ) )
         => topolo1002775350975398744n_open(A,S2) ) ) ).

% openI
tff(fact_7732_open__Collect__conj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),P))
         => ( topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),Q))
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_alk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_7733_open__Collect__disj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),P))
         => ( topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),Q))
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_all(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_7734_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_alm($o,fun(A,$o),(P)))) ) ).

% open_Collect_const
tff(fact_7735_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_7736_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X4) )
           => ~ aa(set(A),$o,member(A,complete_Inf_Inf(A,A3)),A3) ) ) ) ).

% Inf_notin_open
tff(fact_7737_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X) )
           => ~ aa(set(A),$o,member(A,aa(set(A),A,complete_Sup_Sup(A),A3)),A3) ) ) ) ).

% Sup_notin_open
tff(fact_7738_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A),T5: set(A)] :
          ( aa(set(A),$o,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),T5)
             => ( topolo174197925503356063within(A,A2,T5) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_7739_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( aa(set(A),$o,member(A,X),S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,X,B4)),S2) ) ) ) ) ) ).

% open_right
tff(fact_7740_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A)] :
          ( ( topolo174197925503356063within(A,X,S) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_adx(A,A)) = X ) ) ) ).

% Lim_ident_at
tff(fact_7741_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S10: set(A)] :
              ( topolo1002775350975398744n_open(A,S10)
             => ( aa(set(A),$o,member(A,F0),S10)
               => ? [N6: nat] :
                  ! [N2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
                   => aa(set(A),$o,member(A,aa(nat,A,F2,N2)),S10) ) ) ) ) ) ).

% lim_explicit
tff(fact_7742_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_ahz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_7743_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aic(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_7744_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A)))),F3) ) ) ).

% continuous_def
tff(fact_7745_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aid(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_7746_t2__space__class_OLim__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [A3: filter(B),F2: fun(B,A)] : topolo3827282254853284352ce_Lim(B,A,A3,F2) = the(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_aln(filter(B),fun(fun(B,A),fun(A,$o)),A3),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_7747_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S2: set(A),T5: set(A),U3: set(A)] :
          ( aa(set(A),$o,member(A,X),S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T5),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U3),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,X,T5) = topolo174197925503356063within(A,X,U3) ) ) ) ) ) ).

% at_within_nhd
tff(fact_7748_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_ajg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_7749_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_7750_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] :
          ( ( topolo174197925503356063within(A,A2,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_7751_suminf__eq__lim,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_akk(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_7752_lim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X5: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X5) = the(A,aTP_Lamp_alo(fun(nat,A),fun(A,$o),X5)) ) ).

% lim_def
tff(fact_7753_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_alp(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_ahg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_7754_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_alp(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_ahi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_7755_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aii(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_7756_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_alq(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_7757_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_akb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_7758_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(set(real),$o,member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adx(A,A)))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_alr(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_7759_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S2: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( aa(set(B),$o,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_als(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_7760_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F6: fun(A,B),S: set(A),X: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F6)
           => ( ! [Y3: A] :
                  ( aa(set(A),$o,member(A,Y3),S)
                 => ( ( Y3 != X )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,X)),E2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y3)),aa(A,B,F2,X))),aa(A,B,F6,aa(A,A,minus_minus(A,Y3),X))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y3),X)))),aa(A,real,H6,Y3)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S))
               => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_7761_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,X,X) = zero_zero(real) ) ).

% dist_self
tff(fact_7762_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( real_V557655796197034286t_dist(A,X,Y) = zero_zero(real) )
        <=> ( X = Y ) ) ) ).

% dist_eq_0_iff
tff(fact_7763_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,zero_zero(A),X) = real_V7770717601297561774m_norm(A,X) ) ).

% dist_0_norm
tff(fact_7764_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))
        <=> ( X != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_7765_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))
        <=> ( X = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_7766_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,X,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,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),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_7767_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,X,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,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),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_7768_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,D3: real] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_alt(A,fun(real,fun(A,$o)),X),D3))) ) ).

% open_ball
tff(fact_7769_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),S2)
             => ? [E3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
                  & ! [Y5: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y5,X2)),E3)
                     => aa(set(A),$o,member(A,Y5),S2) ) ) ) ) ) ).

% open_dist
tff(fact_7770_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z2: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z2)),real_V557655796197034286t_dist(A,Y,Z2))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_triangle_lt
tff(fact_7771_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_7772_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X)),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_commute_lessI
tff(fact_7773_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)) ) ) ).

% dist_pos_lt
tff(fact_7774_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_7775_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,X) = real_V557655796197034286t_dist(A,X,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_7776_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V557655796197034286t_dist(A,A2,B2)),real_V557655796197034286t_dist(A,B2,C2)))),real_V557655796197034286t_dist(A,A2,C2)) ) ).

% abs_dist_diff_le
tff(fact_7777_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)) ) ).

% zero_le_dist
tff(fact_7778_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z2: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z2)),real_V557655796197034286t_dist(A,Y,Z2))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_triangle_le
tff(fact_7779_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,A2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,X)),real_V557655796197034286t_dist(A,A2,Y))) ) ).

% dist_triangle3
tff(fact_7780_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,Z2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z2)),real_V557655796197034286t_dist(A,Y,Z2))) ) ).

% dist_triangle2
tff(fact_7781_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z2: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Z2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Y)),real_V557655796197034286t_dist(A,Y,Z2))) ) ).

% dist_triangle
tff(fact_7782_continuous__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,B),fun(A,real),aTP_Lamp_alu(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_7783_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,A2: A,S2: set(A),D3: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,A2,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( aa(set(A),$o,member(A,A2),S2)
             => ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D3)
                     => ( aa(A,A,F2,X4) = aa(A,A,G,X4) ) ) )
               => has_field_derivative(A,G,F6,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_7784_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),D3: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( aa(set(A),$o,member(A,X),S)
             => ( ! [X10: A] :
                    ( aa(set(A),$o,member(A,X10),S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D3)
                     => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) )
               => has_derivative(A,B,G,F6,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_7785_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N2))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_7786_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_7787_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,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
               => ! [N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N8)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N8))),E2) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_7788_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_7789_tendsto__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B),M: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),F3)
           => filterlim(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_alv(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(B,L,M)),F3) ) ) ) ).

% tendsto_dist
tff(fact_7790_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),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_7791_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),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_7792_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,minus_minus(B,B2),C2)) = aa(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_7793_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,minus_minus(B,A2),B2)),C2) = aa(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_7794_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,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X15)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2) ) ) ) ).

% dist_triangle_half_r
tff(fact_7795_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,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2) ) ) ) ).

% dist_triangle_half_l
tff(fact_7796_metric__LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,G,X4),M)),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L)) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_7797_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,X: A,S2: set(A),D3: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( ! [X10: A] :
                  ( aa(set(A),$o,member(A,X10),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X10,X))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D3)
                     => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_7798_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,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X42)),E2) ) ) ) ) ).

% dist_triangle_third
tff(fact_7799_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G4: filter(B),X: A,S2: set(A),F3: filter(B),D3: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G4,topolo174197925503356063within(A,X,S2))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G4),F3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
             => ( ! [X10: A] :
                    ( aa(set(A),$o,member(A,X10),S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X10,X))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D3)
                       => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,X,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_7800_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$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_7801_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_7802_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M: nat,Na: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na)))) ) ).

% dist_of_nat
tff(fact_7803_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_alw(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_7804_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_7805_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R3: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X3: A] :
                    ( ( ( X3 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L5)),R3) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_7806_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_7807_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ( ! [X4: A] :
                  ( ( X4 != A2 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),R2)
                   => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_7808_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_7809_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$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)),R) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_7810_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => ? [No: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N8),L5)),R3) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_7811_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Na: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),Na) = 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))),Na)),aa(nat,A,power_power(A,X),Na)) ) ) ) ).

% power_minus'
tff(fact_7812_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M9: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,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_7813_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D2) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alx(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_7814_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D2) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alx(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_7815_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_7816_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_als(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_7817_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [K3: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),K3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_alz(real,fun(A,set(A)),E3)),K3))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_7818_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_ama(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_7819_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),aa(A,set(A),set_ord_greaterThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),I) ) ) ).

% greaterThan_iff
tff(fact_7820_cInf__greaterThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A)
        & no_top(A) )
     => ! [X: A] : complete_Inf_Inf(A,aa(A,set(A),set_ord_greaterThan(A),X)) = X ) ).

% cInf_greaterThan
tff(fact_7821_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% greaterThan_subset_iff
tff(fact_7822_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_7823_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_7824_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A),T5: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S2)
           => topolo6688025880775521714ounded(A,T5) ) ) ) ).

% totally_bounded_subset
tff(fact_7825_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_7826_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
    <=> filterlim(real,A,aTP_Lamp_amb(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),A2)))) ) ).

% filterlim_at_left_to_right
tff(fact_7827_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ? [A4: A,B4: A] :
              ( aa(set(A),$o,member(A,X),aa(A,set(A),set_ord_lessThan(A),A4))
              & aa(set(A),$o,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_7828_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_amc(fun(real,A),fun(real,fun(real,A)),F2),A2),F3,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_7829_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_amd(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_7830_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),aa(real,set(real),set_ord_greaterThan(real),one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_7831_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_ame(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_7832_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,S: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,X,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_amf(A,fun(set(A),fun(A,filter(A))),X),S)),aa(A,set(A),set_ord_greaterThan(A),X)))),complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_amg(A,fun(set(A),fun(A,filter(A))),X),S)),aa(A,set(A),set_ord_lessThan(A),X)))) ) ) ) ).

% at_within_order
tff(fact_7833_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))))) ).

% filterlim_tan_at_right
tff(fact_7834_principal__le__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(A),filter(A),principal(A),A3)),aa(set(A),filter(A),principal(A),B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% principal_le_iff
tff(fact_7835_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),aTP_Lamp_amh(fun(B,set(A)),fun(B,filter(A)),A3)),I5)) = aa(set(A),filter(A),principal(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) ).

% SUP_principal
tff(fact_7836_nhds__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ).

% nhds_discrete
tff(fact_7837_at__bot__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : at_bot(A) = complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_ami(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).

% at_bot_sub
tff(fact_7838_at__bot__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_bot(A) = complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_amj(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_7839_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G4: filter(B),F3: filter(A),P: fun(A,$o),G: fun(A,B)] :
      ( filterlim(A,B,F2,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),P))))
     => ( filterlim(A,B,G,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(fun(A,$o),fun(A,$o),P)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_amk(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G4,F3) ) ) ).

% filterlim_If
tff(fact_7840_nhds__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) = complete_Inf_Inf(filter(A),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),principal(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aml(A,fun(set(A),$o),A2)))) ) ).

% nhds_def
tff(fact_7841_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E4: $tType,D6: $tType,C: $tType,J4: set(A),I: fun(A,C),I5: set(C),F3: fun(C,set(D6)),F2: fun(D6,E4),G4: fun(A,set(E4))] :
      ( ! [M4: A,X4: B] :
          ( aa(set(A),$o,member(A,M4),J4)
         => aa(set(C),$o,member(C,aa(A,C,I,M4)),I5) )
     => ( ! [M4: A,X4: D6] :
            ( aa(set(A),$o,member(A,M4),J4)
           => ( aa(set(D6),$o,member(D6,X4),aa(C,set(D6),F3,aa(A,C,I,M4)))
             => aa(set(E4),$o,member(E4,aa(D6,E4,F2,X4)),aa(A,set(E4),G4,M4)) ) )
       => filterlim(D6,E4,F2,complete_Inf_Inf(filter(E4),aa(set(A),set(filter(E4)),image(A,filter(E4),aTP_Lamp_amm(fun(A,set(E4)),fun(A,filter(E4)),G4)),J4)),complete_Inf_Inf(filter(D6),aa(set(C),set(filter(D6)),image(C,filter(D6),aTP_Lamp_amn(fun(C,set(D6)),fun(C,filter(D6)),F3)),I5))) ) ) ).

% filterlim_base
tff(fact_7842_tendsto__principal__singleton,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),X: A] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ) ).

% tendsto_principal_singleton
tff(fact_7843_nhds__discrete__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A] :
          ( topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))
         => ( topolo7230453075368039082e_nhds(A,X) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ) ) ).

% nhds_discrete_open
tff(fact_7844_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_7845_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_7846_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_7847_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_lessThan(real),zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_7848_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D6: $tType,I5: set(A),F3: fun(A,set(B)),F2: fun(B,C),G4: fun(D6,set(C)),J4: set(D6)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => ! [J2: A] :
                ( aa(set(A),$o,member(A,J2),I5)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,I2)),aa(A,set(B),F3,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,J2)),aa(A,set(B),F3,I2)) ) ) )
       => ( filterlim(B,C,F2,complete_Inf_Inf(filter(C),aa(set(D6),set(filter(C)),image(D6,filter(C),aTP_Lamp_amo(fun(D6,set(C)),fun(D6,filter(C)),G4)),J4)),complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_amp(fun(A,set(B)),fun(A,filter(B)),F3)),I5)))
        <=> ! [X2: D6] :
              ( aa(set(D6),$o,member(D6,X2),J4)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),I5)
                  & ! [Xb4: B] :
                      ( aa(set(B),$o,member(B,Xb4),aa(A,set(B),F3,Xa2))
                     => aa(set(C),$o,member(C,aa(B,C,F2,Xb4)),aa(D6,set(C),G4,X2)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_7849_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_amp(fun(A,set(B)),fun(A,filter(B)),F2)),X5)) = aa(set(B),filter(B),principal(B),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),F2),X5))) ) ) ).

% INF_principal_finite
tff(fact_7850_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_amq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_7851_INT__greaterThan__UNIV,axiom,
    complete_Inf_Inf(set(nat),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_7852_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% ln_at_0
tff(fact_7853_tendsto__at__botI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X7: fun(nat,real)] :
              ( filterlim(nat,real,X7,at_bot(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_amr(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X7),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_bot(real)) ) ) ).

% tendsto_at_botI_sequentially
tff(fact_7854_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)),aa(set(A),filter(A),principal(A),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_7855_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = complete_Inf_Inf(filter(A),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_amt(A,fun(real,filter(A)),X)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_7856_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)) = complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_amu(A,fun(A,filter(A)),X)),aa(A,set(A),set_ord_lessThan(A),X))) ) ) ) ).

% at_left_eq
tff(fact_7857_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)) = complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_amv(A,fun(A,filter(A)),X)),aa(A,set(A),set_ord_greaterThan(A),X))) ) ) ) ).

% at_right_eq
tff(fact_7858_nhds__order,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_amw(A,filter(A))),aa(A,set(A),set_ord_greaterThan(A),X)))),complete_Inf_Inf(filter(A),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_amx(A,filter(A))),aa(A,set(A),set_ord_lessThan(A),X)))) ) ).

% nhds_order
tff(fact_7859_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S: set(A)] : topolo174197925503356063within(A,X,S) = complete_Inf_Inf(filter(A),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_amy(A,fun(set(A),fun(set(A),filter(A))),X),S)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aml(A,fun(set(A),$o),X)))) ) ).

% at_within_eq
tff(fact_7860_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,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_7861_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_amz(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_7862_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_7863_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_amz(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_7864_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,set(real),set_ord_lessThan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% filterlim_tan_at_left
tff(fact_7865_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_7866_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_7867_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_7868_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ana(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_7869_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_amq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_7870_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ana(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_7871_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_7872_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
    <=> filterlim(A,real,aTP_Lamp_anb(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ).

% filterlim_uminus_at_top
tff(fact_7873_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F3)
    <=> filterlim(A,real,aTP_Lamp_anb(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_uminus_at_bot
tff(fact_7874_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_amb(fun(real,A),fun(real,A),F2),F3,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_7875_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_amb(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_7876_filterlim__uminus__at__top__at__bot,axiom,
    filterlim(real,real,uminus_uminus(real),at_top(real),at_bot(real)) ).

% filterlim_uminus_at_top_at_bot
tff(fact_7877_filterlim__uminus__at__bot__at__top,axiom,
    filterlim(real,real,uminus_uminus(real),at_bot(real),at_top(real)) ).

% filterlim_uminus_at_bot_at_top
tff(fact_7878_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(nat,fun(fun(A,real),fun(A,real)),Na),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_7879_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_anc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_7880_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_and(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_7881_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F3: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F3)
    <=> filterlim(A,real,aTP_Lamp_pt(fun(A,nat),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_7882_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_amq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_7883_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ane(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_7884_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F3: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_anf(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_7885_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_ang(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_7886_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_anh(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_7887_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_anh(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_7888_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_7889_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_7890_tendsto__at__topI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X7: fun(nat,real)] :
              ( filterlim(nat,real,X7,at_top(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_amr(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X7),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_7891_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_amq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_7892_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ani(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_7893_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_anj(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_7894_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_7895_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X4)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X4,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_7896_at__infinity__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( at_infinity(A) = complete_Inf_Inf(filter(A),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_anl(real,filter(A))),top_top(set(real)))) ) ) ).

% at_infinity_def
tff(fact_7897_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G3: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
     => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_7898_eventually__const,axiom,
    ! [A: $tType,F3: filter(A),P: $o] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_yf($o,fun(A,$o),(P)),F3)
      <=> (P) ) ) ).

% eventually_const
tff(fact_7899_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
          <=> ? [B11: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B11)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B11)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_7900_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
        <=> ? [B11: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B11)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B11)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_7901_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),X),topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X))) ) ).

% eventually_at_right_less
tff(fact_7902_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_anp(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7903_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_7904_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_anq(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_7905_at__bot__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_bot(real)),at_infinity(real)) ).

% at_bot_le_at_infinity
tff(fact_7906_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_7907_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_tw(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_7908_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_anr(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_bot
tff(fact_7909_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ans(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_top
tff(fact_7910_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_anr(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7911_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ans(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7912_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ant(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,at_top(B),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7913_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,$o),G: fun(B,A)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7914_eventually__nhds__within__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A2: A,S: set(A)] :
          ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),aa(set(A),filter(A),principal(A),S)))
        <=> ! [F7: fun(nat,A)] :
              ( ( ! [N2: nat] : aa(set(A),$o,member(A,aa(nat,A,F7,N2)),S)
                & filterlim(nat,A,F7,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F7),at_top(nat)) ) ) ) ).

% eventually_nhds_within_iff_sequentially
tff(fact_7915_sequentially__imp__eventually__nhds__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S: set(A),A2: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(set(A),$o,member(A,aa(nat,A,F4,N8)),S)
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),aa(set(A),filter(A),principal(A),S))) ) ) ).

% sequentially_imp_eventually_nhds_within
tff(fact_7916_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ! [F7: fun(nat,A)] :
              ( filterlim(nat,A,F7,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F7),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_7917_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_anv(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_7918_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_anw(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_7919_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_anx(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_7920_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_any(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_7921_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A2: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(nat,A,F4,N8) != A2
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_7922_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S: set(A),A2: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N8: nat] :
                    ( aa(set(A),$o,member(A,aa(nat,A,F4,N8)),S)
                    & ( aa(nat,A,F4,N8) != A2 ) )
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,S)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_7923_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I: A,I5: set(A),P: fun(B,$o),F3: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,I),I5)
     => ( eventually(B,P,aa(A,filter(B),F3,I))
       => eventually(B,P,complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),I5))) ) ) ).

% eventually_INF1
tff(fact_7924_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_7925_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_7926_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_anz(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_7927_at__top__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_top(real)),at_infinity(real)) ).

% at_top_le_at_infinity
tff(fact_7928_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_7929_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X4)
             => aa(A,$o,P,X4) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7930_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_7931_eventually__Inf__base,axiom,
    ! [A: $tType,B3: set(filter(A)),P: fun(A,$o)] :
      ( ( B3 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( aa(set(filter(A)),$o,member(filter(A),F5),B3)
           => ! [G2: filter(A)] :
                ( aa(set(filter(A)),$o,member(filter(A),G2),B3)
               => ? [X3: filter(A)] :
                    ( aa(set(filter(A)),$o,member(filter(A),X3),B3)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X3),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G2)) ) ) )
       => ( eventually(A,P,complete_Inf_Inf(filter(A),B3))
        <=> ? [X2: filter(A)] :
              ( aa(set(filter(A)),$o,member(filter(A),X2),B3)
              & eventually(A,P,X2) ) ) ) ) ).

% eventually_Inf_base
tff(fact_7932_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( aa(set(A),$o,member(A,X),S)
           => eventually(A,aTP_Lamp_aoa(set(A),fun(A,$o),S),topolo7230453075368039082e_nhds(A,X)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_7933_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_aob(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),A2),S),topolo7230453075368039082e_nhds(A,A2)) ) ) ).

% eventually_at_filter
tff(fact_7934_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,X,S))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,F2,X) = aa(A,B,G,X) )
             => ( aa(set(A),$o,member(A,X),S)
               => has_derivative(A,B,G,F6,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_7935_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)))
          <=> ? [B11: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),top_top(A))
                & ! [Z6: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),Z6)
                   => aa(A,$o,P,Z6) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_7936_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,aTP_Lamp_aod(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,X))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,X)) ) ) ).

% eventually_eventually
tff(fact_7937_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => eventually(A,aTP_Lamp_aoe(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,X)) ) ) ).

% t1_space_nhds
tff(fact_7938_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),X: A,S2: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_aof(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,A,F2,X) = aa(A,A,G,X) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S2))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_7939_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,$o)] :
          ( eventually(A,P3,at_infinity(A))
        <=> ? [B11: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B11)
              & ! [X2: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B11),real_V7770717601297561774m_norm(A,X2))
                 => aa(A,$o,P3,X2) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7940_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( ~ eventually(A,Q,F3)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% not_eventually_impI
tff(fact_7941_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
    <=> ( eventually(A,P,F3)
        & eventually(A,Q,F3) ) ) ).

% eventually_conj_iff
tff(fact_7942_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_rev_mp
tff(fact_7943_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aog(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
      <=> eventually(A,Q,F3) ) ) ).

% eventually_subst
tff(fact_7944_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o),R2: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => ( ! [I2: A] :
              ( aa(A,$o,P,I2)
             => ( aa(A,$o,Q,I2)
               => aa(A,$o,R2,I2) ) )
         => eventually(A,R2,F3) ) ) ) ).

% eventually_elim2
tff(fact_7945_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% eventually_conj
tff(fact_7946_eventually__True,axiom,
    ! [A: $tType,F3: filter(A)] : eventually(A,aTP_Lamp_rc(A,$o),F3) ).

% eventually_True
tff(fact_7947_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_mp
tff(fact_7948_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F3: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_aoh(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F3)
    <=> ( eventually(A,P,F3)
        | (C3) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7949_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aoi($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
        | eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7950_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aoj($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
       => eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7951_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X4: A] : ~ aa(A,$o,P,X4)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7952_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F3: filter(A)] :
      ( eventually(A,aTP_Lamp_yf($o,fun(A,$o),(P)),F3)
    <=> ( (P)
        | ( F3 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_7953_trivial__limit__def,axiom,
    ! [A: $tType,F3: filter(A)] :
      ( ( F3 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_dc(A,$o),F3) ) ).

% trivial_limit_def
tff(fact_7954_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : eventually(A,aTP_Lamp_aok(A,fun(A,$o),A2),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_7955_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B11: real] :
            ! [X2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B11),real_V7770717601297561774m_norm(A,X2))
             => aa(A,$o,P,X2) ) ) ) ).

% eventually_at_infinity
tff(fact_7956_filter__leD,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A),P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F9)
     => ( eventually(A,P,F9)
       => eventually(A,P,F3) ) ) ).

% filter_leD
tff(fact_7957_filter__leI,axiom,
    ! [A: $tType,F9: filter(A),F3: filter(A)] :
      ( ! [P5: fun(A,$o)] :
          ( eventually(A,P5,F9)
         => eventually(A,P5,F3) )
     => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F9) ) ).

% filter_leI
tff(fact_7958_le__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F9)
    <=> ! [P6: fun(A,$o)] :
          ( eventually(A,P6,F9)
         => eventually(A,P6,F3) ) ) ).

% le_filter_def
tff(fact_7959_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,S2: set(A),F2: fun(A,A),G: fun(A,A),U: A,V2: A,T2: set(A)] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_aol(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),S2),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V2 )
             => ( ( S2 = T2 )
               => ( aa(set(A),$o,member(A,X),S2)
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S2))
                  <=> has_field_derivative(A,G,V2,topolo174197925503356063within(A,Y,T2)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_7960_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,$o)),X: A,X5: set(A)] :
          ( eventually(A,aa(A,fun(A,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,X5),aTP_Lamp_adx(A,A))),topolo174197925503356063within(A,X,X5))
        <=> eventually(A,aa(A,fun(A,$o),P,X),topolo174197925503356063within(A,X,X5)) ) ) ).

% eventually_Lim_ident_at
tff(fact_7961_real__tendsto__sandwich,axiom,
    ! [A: $tType,F2: fun(A,real),G: fun(A,real),Net: filter(A),H: fun(A,real),C2: real] :
      ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_aom(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_aom(fun(A,real),fun(fun(A,real),fun(A,$o)),G),H),Net)
       => ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(A,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_7962_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),A2: B,S: set(B),F3: filter(A),B2: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,A2,S),F3)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_aon(fun(A,B),fun(set(B),fun(B,fun(A,$o))),F2),S),B2),F3) ) ) ).

% filterlim_at_within_not_equal
tff(fact_7963_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F3: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F3)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_aoo(fun(A,B),fun(real,fun(A,$o)),F2),R5),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7964_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: filter(A),F2: fun(B,A),G4: filter(B)] :
      ( eventually(A,P,F3)
     => ( filterlim(B,A,F2,F3,G4)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aop(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),G4) ) ) ).

% eventually_compose_filterlim
tff(fact_7965_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),S2: set(B),F3: filter(A)] :
      ( filterlim(A,B,F2,aa(set(B),filter(B),principal(B),S2),F3)
    <=> eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_aoq(fun(A,B),fun(set(B),fun(A,$o)),F2),S2),F3) ) ).

% filterlim_principal
tff(fact_7966_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aor(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G),F23)
         => ( filterlim(B,A,F2,F12,F23)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7967_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
    <=> ! [P6: fun(B,$o)] :
          ( eventually(B,P6,F23)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aos(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P6),F12) ) ) ).

% filterlim_iff
tff(fact_7968_tendsto__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,Net: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aot(fun(A,B),fun(B,fun(A,$o)),F2),L),Net)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_7969_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),C9: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != C9 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aou(fun(A,B),fun(B,fun(A,$o)),F2),C9),F3) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_7970_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_aov(fun(A,B),fun(B,fun(A,$o)),F2),Y),F3) ) ) ).

% tendsto_discrete
tff(fact_7971_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aow(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3) ) ) ) ).

% tendsto_cong
tff(fact_7972_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aox(fun(A,B),fun(B,fun(A,$o)),F2),C2),F3) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_7973_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aow(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% Lim_transform_eventually
tff(fact_7974_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G4: filter(A),F9: filter(B),G5: filter(A),F6: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G4)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F3),F9)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),G4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoy(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F6),G5)
           => filterlim(A,B,F6,F9,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7975_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_aoz(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoz(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_7976_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_7977_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_7978_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_apc(fun(B,A),fun(A,fun(B,$o)),F2),A4),F3) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_apd(fun(B,A),fun(A,fun(B,$o)),F2),A4),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3) ) ) ) ).

% order_tendstoI
tff(fact_7979_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,$o)),F2),L4),F3) )
            & ! [U4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),U4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(B,fun(A,$o)),F2),U4),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_7980_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),B2: B,S: set(B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,B2,S),F3)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ape(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),B2),S),F3)
            & filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),F3) ) ) ) ).

% filterlim_at
tff(fact_7981_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),S: set(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),aa(set(A),filter(A),principal(A),S)))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_apf(fun(A,$o),fun(set(A),fun(A,$o)),P),S),F3) ) ).

% eventually_inf_principal
tff(fact_7982_le__principal,axiom,
    ! [A: $tType,F3: filter(A),A3: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),aa(set(A),filter(A),principal(A),A3))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),A3),F3) ) ).

% le_principal
tff(fact_7983_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)))
        <=> ? [B11: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),X)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_7984_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)))
          <=> ? [B11: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),X)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B11),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_7985_tendsto__def,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [S10: set(B)] :
              ( topolo1002775350975398744n_open(B,S10)
             => ( aa(set(B),$o,member(B,L),S10)
               => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_apg(fun(A,B),fun(set(B),fun(A,$o)),F2),S10),F3) ) ) ) ) ).

% tendsto_def
tff(fact_7986_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),S2: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( topolo1002775350975398744n_open(B,S2)
           => ( aa(set(B),$o,member(B,L),S2)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_apg(fun(A,B),fun(set(B),fun(A,$o)),F2),S2),F3) ) ) ) ) ).

% topological_tendstoD
tff(fact_7987_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [S4: set(A)] :
              ( topolo1002775350975398744n_open(A,S4)
             => ( aa(set(A),$o,member(A,L),S4)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_aph(fun(B,A),fun(set(A),fun(B,$o)),F2),S4),F3) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% topological_tendstoI
tff(fact_7988_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,$o),F3: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( eventually(B,P,complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),A3)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => eventually(B,aa(A,fun(B,$o),Q7,X2),aa(A,filter(B),F3,X2)) )
            & ! [Y5: B] :
                ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X2),Y5) )
               => aa(B,$o,P,Y5) ) ) ) ) ).

% eventually_INF_finite
tff(fact_7989_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))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S2)
                 => ( ( ( X2 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D4) )
                   => aa(A,$o,P,X2) ) ) ) ) ) ).

% eventually_at
tff(fact_7990_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X2: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D4)
                 => aa(A,$o,P,X2) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_7991_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X4) )
         => ( 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_7992_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X4) )
         => ( 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_7993_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,$o),aTP_Lamp_api(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_7994_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F3: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_apj(B,fun(fun(A,B),fun(A,$o)),L),F2),F3)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(B,fun(A,$o)),F2),X4),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_7995_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apk(fun(A,B),fun(B,fun(A,$o)),F2),L),F3)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,$o)),F2),X4),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_7996_filterlim__real__at__infinity__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_infinity(real),at_top(nat)) ).

% filterlim_real_at_infinity_sequentially
tff(fact_7997_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apl(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7998_filterlim__atI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apm(fun(A,B),fun(B,fun(A,$o)),F2),C2),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,top_top(set(B))),F3) ) ) ) ).

% filterlim_atI
tff(fact_7999_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apn(fun(A,B),fun(B,fun(A,$o)),F2),B2),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_apo(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose_eventually
tff(fact_8000_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),M: B,L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_app(A,fun(fun(C,A),fun(C,$o)),L),F2),F3)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,M),F3) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_8001_tendsto__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => filterlim(nat,A,semiring_1_of_nat(A),at_infinity(A),at_top(nat)) ) ).

% tendsto_of_nat
tff(fact_8002_isCont__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_apq(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,X))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
          <=> topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),G) ) ) ) ).

% isCont_cong
tff(fact_8003_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,F2: fun(A,A),G: fun(A,A),U: A,V2: A] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_aof(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V2 )
             => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,top_top(set(A))))
              <=> has_field_derivative(A,G,V2,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_8004_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apr(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_8005_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aps(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_8006_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apt(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_8007_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: filter(A),F2: fun(A,B),X: B,G: fun(A,B),Y: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F3)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_apu(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ) ) ) ).

% tendsto_le
tff(fact_8008_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( eventually(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_apv(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F2),A2),G),B2),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F3) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_8009_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apw(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_floor_eq
tff(fact_8010_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_apx(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_ceiling_eq
tff(fact_8011_eventually__at__right__to__0,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
    <=> eventually(real,aa(real,fun(real,$o),aTP_Lamp_apy(fun(real,$o),fun(real,fun(real,$o)),P),A2),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_8012_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,filter(A)),B3: set(B)] :
      ( eventually(A,P,complete_Inf_Inf(filter(A),aa(set(B),set(filter(A)),image(B,filter(A),F3),B3)))
    <=> ? [X9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B3)
          & aa(set(B),$o,finite_finite2(B),X9)
          & eventually(A,P,complete_Inf_Inf(filter(A),aa(set(B),set(filter(A)),image(B,filter(A),F3),X9))) ) ) ).

% eventually_INF
tff(fact_8013_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_apz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_8014_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_apz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_8015_eventually__at__left__to__right,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
    <=> eventually(real,aTP_Lamp_aqa(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),A2)))) ) ).

% eventually_at_left_to_right
tff(fact_8016_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( eventually(A,aTP_Lamp_aqb(fun(A,real),fun(A,$o),F2),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_alq(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_8017_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,$o),aTP_Lamp_aqc(real,fun(real,fun(real,$o)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_8018_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,$o),aTP_Lamp_aqc(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ).

% eventually_at_left_real
tff(fact_8019_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))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S2)
                 => ( ( ( X2 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X2,A2)),D4) )
                   => aa(A,$o,P,X2) ) ) ) ) ) ).

% eventually_at_le
tff(fact_8020_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Na: nat,B3: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_aqd(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Na),B3),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_8021_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Y: B,F3: filter(A),G: fun(B,C),Z2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z2),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_aqe(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),F2),Y),G),Z2),F3)
             => filterlim(A,C,comp(B,C,A,G,F2),topolo7230453075368039082e_nhds(C,Z2),F3) ) ) ) ) ).

% tendsto_compose_at
tff(fact_8022_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_lessThan(B),L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_8023_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_8024_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),E2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqf(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F3) ) ) ) ).

% tendstoD
tff(fact_8025_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqf(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F3) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% tendstoI
tff(fact_8026_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqf(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3),F3) ) ) ) ).

% tendsto_iff
tff(fact_8027_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,$o),B3: set(filter(A))] :
      ( eventually(A,P,complete_Inf_Inf(filter(A),B3))
    <=> ? [X9: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X9),B3)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X9)
          & eventually(A,P,complete_Inf_Inf(filter(A),X9)) ) ) ).

% eventually_Inf
tff(fact_8028_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_8029_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
    <=> eventually(real,aTP_Lamp_aqg(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_8030_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_aqg(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_8031_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_aqh(fun(A,real),fun(A,$o),F2),F3)
         => filterlim(A,real,aTP_Lamp_aev(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_8032_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,$o),G: fun(B,A),A2: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( 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_8033_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F3)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_8034_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_8035_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_aqk(fun(A,B),fun(nat,fun(A,B)),F2),Na),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_8036_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F3: fun(A,filter(B)),P: fun(B,$o)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),B3)
           => ! [B4: A] :
                ( aa(set(A),$o,member(A,B4),B3)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),B3)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F3,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,A4)),aa(A,filter(B),F3,B4))) ) ) )
       => ( eventually(B,P,complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F3),B3)))
        <=> ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),B3)
              & eventually(B,P,aa(A,filter(B),F3,X2)) ) ) ) ) ).

% eventually_INF_base
tff(fact_8037_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,$o),G: fun(B,A),A2: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( 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_8038_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G4: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),G4)
        <=> filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),at_top(real),G4) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_8039_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),at_top(real),F3)
         => filterlim(A,B,F2,at_infinity(B),F3) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_8040_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => filterlim(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_8041_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K6: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_aql(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K6),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_8042_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),A3: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aqm(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C2),A3),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,A3),F3) ) ) ) ).

% filterlim_at_withinI
tff(fact_8043_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_aqn(fun(A,real),fun(A,$o),F2),F3)
         => ( 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_ahf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_8044_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_aqn(fun(A,real),fun(A,$o),F2),F3)
         => ( 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_ahf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_8045_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_aqn(fun(A,real),fun(A,$o),F2),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_8046_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aqo(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_floor_less
tff(fact_8047_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aqp(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_less_ceiling
tff(fact_8048_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( eventually(A,aTP_Lamp_any(fun(A,real),fun(A,$o),G),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_anc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_8049_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_any(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,F2,at_top(real),F3)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_8050_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_any(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,aTP_Lamp_and(fun(A,real),fun(A,real),F2),at_top(real),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_8051_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_any(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_and(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_8052_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_anx(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_and(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_8053_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_8054_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_8055_lim__infinity__imp__sequentially,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(real,A),L: A] :
          ( filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(real))
         => filterlim(nat,A,aTP_Lamp_aqr(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_8056_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_ahd(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F3)
        <=> filterlim(A,B,G,at_infinity(B),F3) ) ) ).

% filterlim_inverse_at_iff
tff(fact_8057_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G3: fun(real,real),F6: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),F3,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_8058_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_8059_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_8060_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_8061_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aad(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_8062_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G3: fun(real,real),F2: fun(real,real),F6: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_8063_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G3: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_8064_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G3: fun(real,real),F6: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
       => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G0),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
         => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F0),F6),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G0),G3),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),F3,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F3,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_8065_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G3: fun(real,real),F6: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
       => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
         => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),F3,topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_8066_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X))
         => filterlim(nat,A,power_power(A,X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_8067_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G3: fun(real,real),F6: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
       => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
         => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),F3,topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,aa(real,set(real),set_ord_lessThan(real),X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_8068_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_8069_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G3: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G3),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_8070_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,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_8071_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_aqs(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_8072_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G3: fun(real,real),F2: fun(real,real),F6: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
     => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_8073_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G3: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
     => ( eventually(real,aTP_Lamp_anm(fun(real,real),fun(real,$o),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G3),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,aa(real,set(real),set_ord_greaterThan(real),X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_8074_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y5: B,K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqt(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y5),K7),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_8075_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B5)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_aqu(fun(A,B),fun(real,fun(A,$o)),F2),B5),F3) ) ) ) ).

% BfunE
tff(fact_8076_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_rt(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_8077_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aqv(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_8078_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_sd(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_8079_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aqv(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_8080_summable__cong,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aqw(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(A,F2)
          <=> summable(A,G) ) ) ) ).

% summable_cong
tff(fact_8081_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_aqx(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),F2),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_8082_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_8083_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X4)
         => aa(nat,$o,P,X4) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_8084_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F3),at_top(nat))
    <=> ! [N6: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),F3) ) ).

% le_sequentially
tff(fact_8085_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A8: fun(nat,set(A))] :
              ( ! [I3: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A8,I3))
             => ( ! [I3: nat] : aa(set(A),$o,member(A,X),aa(nat,set(A),A8,I3))
               => ~ ! [S9: set(A)] :
                      ( topolo1002775350975398744n_open(A,S9)
                     => ( aa(set(A),$o,member(A,X),S9)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_aqy(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A8),S9),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_8086_Bfun__const,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [C2: B,F3: filter(A)] : bfun(A,B,aTP_Lamp_aqz(B,fun(A,B),C2),F3) ) ).

% Bfun_const
tff(fact_8087_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),X5),at_top(nat))
        <=> bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_8088_Bseq__subseq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ara(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat)) ) ) ).

% Bseq_subseq
tff(fact_8089_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_8090_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_arb(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_8091_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_arb(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_8092_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_arc(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_8093_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_arc(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_8094_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ard(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_8095_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),K6: real] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K6)
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% BseqI'
tff(fact_8096_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_8097_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_ar(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_8098_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_are(fun(int,A),fun(nat,A),F2),F3,at_top(nat))
     => filterlim(int,A,F2,F3,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_8099_Bseq__eq__bounded,axiom,
    ! [F2: fun(nat,real),A2: real,B2: real] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),aa(set(nat),set(real),image(nat,real,F2),top_top(set(nat)))),set_or1337092689740270186AtMost(real,A2,B2))
     => bfun(nat,real,F2,at_top(nat)) ) ).

% Bseq_eq_bounded
tff(fact_8100_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_arf(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_8101_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,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N8))),K8) ) ) ) ).

% BseqD
tff(fact_8102_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,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N8))),K8) ) ) ) ).

% BseqE
tff(fact_8103_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K6: real,X5: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K6)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_8104_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),K7) ) ) ) ).

% Bseq_def
tff(fact_8105_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_8106_Bseq__iff,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_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff
tff(fact_8107_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => bfun(nat,real,power_power(real,X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_8108_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K6: real,F3: filter(A)] :
          ( eventually(A,aa(real,fun(A,$o),aTP_Lamp_aqu(fun(A,B),fun(real,fun(A,$o)),F2),K6),F3)
         => bfun(A,B,F2,F3) ) ) ).

% BfunI
tff(fact_8109_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X2: A] :
                ! [N2: nat] : aa(real,$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)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_8110_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N6: nat] :
                ! [N2: nat] : aa(real,$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))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_8111_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_ahd(fun(A,B),fun(A,B),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_8112_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_aqu(fun(A,B),fun(real,fun(A,$o)),F2),K7),F3) ) ) ) ).

% Bfun_def
tff(fact_8113_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_arg(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_8114_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)
         => ( ! [F4: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F4,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F4,N8)),B2)
                 => ( order_antimono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_arh(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),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_8115_decseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_antimono(nat,A,aTP_Lamp_ari(A,fun(nat,A),K)) ) ).

% decseq_const
tff(fact_8116_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_arj(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_8117_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y3: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_ark(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y3),Net)
         => eventually(B,aTP_Lamp_arl(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_8118_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,I)) ) ) ) ).

% decseqD
tff(fact_8119_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_8120_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoD
tff(fact_8121_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoE
tff(fact_8122_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X4)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_8123_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X2: A,Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y5)),aa(A,B,F2,X2)) ) ) ) ).

% antimono_def
tff(fact_8124_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_8125_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_8126_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,I))),aa(nat,A,A3,I)) ) ) ).

% decseq_SucD
tff(fact_8127_decseq__bounded,axiom,
    ! [X5: fun(nat,real),B3: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),aa(nat,real,X5,I2))
       => bfun(nat,real,X5,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_8128_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_arm(fun(A,fun(B,$o)),fun(A,$o),P)) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_xg(fun(A,fun(B,$o)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_8129_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),D3: B] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_arn(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A3),B3),D3))) ) ) ).

% finite_set_of_finite_funs
tff(fact_8130_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_aro(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_8131_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_arp(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_8132_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Na: nat] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L5),aa(nat,A,X5,Na)) ) ) ) ).

% decseq_ge
tff(fact_8133_decseq__convergent,axiom,
    ! [X5: fun(nat,real),B3: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),aa(nat,real,X5,I2))
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(nat,real,X5,I3)) ) ) ) ).

% decseq_convergent
tff(fact_8134_nhds__countable,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [X7: fun(nat,set(A))] :
              ( order_antimono(nat,set(A),X7)
             => ( ! [N8: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),X7,N8))
               => ( ! [N8: nat] : aa(set(A),$o,member(A,X),aa(nat,set(A),X7,N8))
                 => ( topolo7230453075368039082e_nhds(A,X) != complete_Inf_Inf(filter(A),aa(set(nat),set(filter(A)),image(nat,filter(A),aTP_Lamp_arq(fun(nat,set(A)),fun(nat,filter(A)),X7)),top_top(set(nat)))) ) ) ) ) ) ).

% nhds_countable
tff(fact_8135_LIMSEQ__INF,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,complete_Inf_Inf(A,aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))),at_top(nat)) ) ) ).

% LIMSEQ_INF
tff(fact_8136_INF__Lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A),L: A] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L),at_top(nat))
           => ( complete_Inf_Inf(A,aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat)))) = L ) ) ) ) ).

% INF_Lim
tff(fact_8137_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,$o),ord_less(A),A2),B2)
         => ( ! [S4: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,S4,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S4,N8)),B2)
                 => ( order_antimono(nat,A,S4)
                   => ( filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_arr(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S4),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_8138_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_ars(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_8139_max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : max_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_art(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R2))) ).

% max_ext_eq
tff(fact_8140_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_8141_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_8142_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_8143_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_8144_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_8145_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_8146_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_8147_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( complete_Inf_Inf(A,set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_8148_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( complete_Inf_Inf(A,set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_8149_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_8150_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D3) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2) )
            | ( ( A2 = C2 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_8151_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_8152_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_8153_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_8154_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_8155_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( aa(set(A),$o,member(A,X),S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,X)),S2) ) ) ) ) ) ).

% open_left
tff(fact_8156_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_8157_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or3652927894154168847AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_8158_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_8159_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% sum.head
tff(fact_8160_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% prod.head
tff(fact_8161_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_8162_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_8163_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_8164_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_8165_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_8166_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_8167_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_8168_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_8169_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_8170_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_8171_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_8172_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X3: set(A),Xa3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_aru(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X3),Xa3)
    <=> aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X3),Xa3)),max_ext(A,R2)) ) ).

% max_extp_max_ext_eq
tff(fact_8173_max__extp__eq,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),X: set(A),Y: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),X),Y)
    <=> aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y)),max_ext(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R3)))) ) ).

% max_extp_eq
tff(fact_8174_max__ext__def,axiom,
    ! [A: $tType,X3: set(product_prod(A,A))] : max_ext(A,X3) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_aru(set(product_prod(A,A)),fun(A,fun(A,$o)),X3)))) ).

% max_ext_def
tff(fact_8175_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_8176_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A)] :
          ( ! [A4: A,B4: A,X4: A] :
              ( aa(set(A),$o,member(A,A4),S2)
             => ( aa(set(A),$o,member(A,B4),S2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B4)
                   => aa(set(A),$o,member(A,X4),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 = 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_8177_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y4: set(B),X5: set(A),F3: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),Y4)),X5)
     => ( eventually(B,aTP_Lamp_arv(set(B),fun(B,$o),Y4),F3)
       => ( map_filter_on(A,C,X5,F2,map_filter_on(B,A,Y4,G,F3)) = map_filter_on(B,C,Y4,comp(A,C,B,F2,G),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_8178_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_8179_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),set_ord_atLeast(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),I) ) ) ).

% atLeast_iff
tff(fact_8180_ATP_Olambda__1,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_bn(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uu2)),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),Uu2),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) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_aig(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu2)),one_one(A))),Uu2) ) ).

% ATP.lambda_2
tff(fact_8182_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu2: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_ut(set(set(A)),int),Uu2) = 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)),Uu2)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),complete_Inf_Inf(set(A),Uu2)))) ).

% ATP.lambda_3
tff(fact_8183_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_za(A,set(product_prod(A,A))),Uu2) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8184_ATP_Olambda__5,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ce(nat,real),Uu2) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_5
tff(fact_8185_ATP_Olambda__6,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_cx(real,$o),Uu2)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu2),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu2) = zero_zero(real) ) ) ) ).

% ATP.lambda_6
tff(fact_8186_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_akd(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_7
tff(fact_8187_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_gz(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu2)),aa(nat,real,power_power(real,zero_zero(real)),Uu2)) ).

% ATP.lambda_8
tff(fact_8188_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_ake(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))) ) ).

% ATP.lambda_9
tff(fact_8189_ATP_Olambda__10,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_aik(real,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu2)),sin(real,Uu2)) ).

% ATP.lambda_10
tff(fact_8190_ATP_Olambda__11,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_ang(real,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu2)),Uu2) ).

% ATP.lambda_11
tff(fact_8191_ATP_Olambda__12,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ajl(nat,real),Uu2) = aa(real,real,root(Uu2),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_12
tff(fact_8192_ATP_Olambda__13,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_no(nat,nat),Uu2) = aa(nat,nat,minus_minus(nat,Uu2),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_13
tff(fact_8193_ATP_Olambda__14,axiom,
    ! [B: $tType,Uu2: B] : aa(B,product_prod(B,B),aTP_Lamp_uj(B,product_prod(B,B)),Uu2) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu2),Uu2) ).

% ATP.lambda_14
tff(fact_8194_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu2: A] : aa(A,product_prod(A,A),aTP_Lamp_uk(A,product_prod(A,A)),Uu2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2) ).

% ATP.lambda_15
tff(fact_8195_ATP_Olambda__16,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ah(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),one_one(A)) ) ).

% ATP.lambda_16
tff(fact_8196_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu2: A] : aa(A,list(A),aTP_Lamp_so(A,list(A)),Uu2) = aa(list(A),list(A),cons(A,Uu2),nil(A)) ).

% ATP.lambda_17
tff(fact_8197_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(A),aTP_Lamp_wa(A,set(A)),Uu2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))) ).

% ATP.lambda_18
tff(fact_8198_ATP_Olambda__19,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ajv(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_19
tff(fact_8199_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_akc(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_20
tff(fact_8200_ATP_Olambda__21,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_th(list(B),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,minus_minus(nat,aa(list(B),nat,size_size(list(B)),Uu2)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_21
tff(fact_8201_ATP_Olambda__22,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_yw(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu2) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2)) ).

% ATP.lambda_22
tff(fact_8202_ATP_Olambda__23,axiom,
    ! [B: $tType,Uu2: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ti(list(B),$o),Uu2)
    <=> ( Uu2 != nil(B) ) ) ).

% ATP.lambda_23
tff(fact_8203_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu2: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_tj(list(A),$o),Uu2)
    <=> ( Uu2 != nil(A) ) ) ).

% ATP.lambda_24
tff(fact_8204_ATP_Olambda__25,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_aa(nat,nat),Uu2) = order_Greatest(nat,aTP_Lamp_a(nat,fun(nat,$o),Uu2)) ).

% ATP.lambda_25
tff(fact_8205_ATP_Olambda__26,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_abd(real,real),Uu2) = suminf(real,aTP_Lamp_ap(real,fun(nat,real),Uu2)) ).

% ATP.lambda_26
tff(fact_8206_ATP_Olambda__27,axiom,
    ! [Uu2: nat] : aa(nat,set(nat),aTP_Lamp_zl(nat,set(nat)),Uu2) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ld(nat,fun(nat,$o),Uu2)) ).

% ATP.lambda_27
tff(fact_8207_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real] : aa(real,filter(A),aTP_Lamp_anl(real,filter(A)),Uu2) = aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ank(real,fun(A,$o),Uu2))) ) ).

% ATP.lambda_28
tff(fact_8208_ATP_Olambda__29,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ajx(nat,real),Uu2) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu2))) ).

% ATP.lambda_29
tff(fact_8209_ATP_Olambda__30,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_dh(nat,A),Uu2) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu2)) ) ).

% ATP.lambda_30
tff(fact_8210_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_ajp(nat,A),Uu2) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_31
tff(fact_8211_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_tg(list(B),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu2)) ).

% ATP.lambda_32
tff(fact_8212_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu2: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_sr(list(A),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu2)) ).

% ATP.lambda_33
tff(fact_8213_ATP_Olambda__34,axiom,
    ! [Uu2: num] : aa(num,option(num),aTP_Lamp_rj(num,option(num)),Uu2) = some(num,aa(num,num,bit1,Uu2)) ).

% ATP.lambda_34
tff(fact_8214_ATP_Olambda__35,axiom,
    ! [Uu2: num] : aa(num,option(num),aTP_Lamp_ri(num,option(num)),Uu2) = some(num,bit0(Uu2)) ).

% ATP.lambda_35
tff(fact_8215_ATP_Olambda__36,axiom,
    ! [Uu2: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pe(int,fun(int,product_prod(int,int))),Uu2) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu2)) ).

% ATP.lambda_36
tff(fact_8216_ATP_Olambda__37,axiom,
    ! [Uu2: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ph(int,fun(int,product_prod(int,int))),Uu2) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu2)) ).

% ATP.lambda_37
tff(fact_8217_ATP_Olambda__38,axiom,
    ! [Uu2: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_oi(nat,fun(nat,product_prod(nat,nat))),Uu2) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_38
tff(fact_8218_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: A] : aa(A,filter(A),aTP_Lamp_amx(A,filter(A)),Uu2) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_greaterThan(A),Uu2)) ) ).

% ATP.lambda_39
tff(fact_8219_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: A] : aa(A,filter(A),aTP_Lamp_amw(A,filter(A)),Uu2) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_lessThan(A),Uu2)) ) ).

% ATP.lambda_40
tff(fact_8220_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,filter(A),aTP_Lamp_ami(A,filter(A)),Uu2) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu2)) ) ).

% ATP.lambda_41
tff(fact_8221_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: A] : aa(A,filter(A),aTP_Lamp_amj(A,filter(A)),Uu2) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu2)) ) ).

% ATP.lambda_42
tff(fact_8222_ATP_Olambda__43,axiom,
    ! [Uu2: int] : aa(int,nat,aTP_Lamp_zm(int,nat),Uu2) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu2)) ).

% ATP.lambda_43
tff(fact_8223_ATP_Olambda__44,axiom,
    ! [Uu2: nat] : aa(nat,option(num),aTP_Lamp_rx(nat,option(num)),Uu2) = some(num,one2) ).

% ATP.lambda_44
tff(fact_8224_ATP_Olambda__45,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_sb(num,fun(nat,option(num)),Uu2),Uua) = case_num(option(num),some(num,one2),aTP_Lamp_rz(nat,fun(num,option(num)),Uua),aTP_Lamp_sa(nat,fun(num,option(num)),Uua),Uu2) ).

% ATP.lambda_45
tff(fact_8225_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_dw(A,fun(nat,A),Uu2),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,Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_46
tff(fact_8226_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_fy(nat,fun(nat,A),Uu2),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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_47
tff(fact_8227_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_dv(A,fun(nat,A),Uu2),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,Uu2),Uua))) ) ).

% ATP.lambda_48
tff(fact_8228_ATP_Olambda__49,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_cf(fun(nat,real),fun(nat,real),Uu2),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uua),zero_zero(real),aa(nat,real,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_49
tff(fact_8229_ATP_Olambda__50,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat] :
      aa(nat,nat,aTP_Lamp_rk(vEBT_VEBT,fun(nat,nat),Uu2),Uua) = $ite(aa(nat,$o,vEBT_vebt_member(Uu2),Uua),Uua,aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Uu2,Uua))) ).

% ATP.lambda_50
tff(fact_8230_ATP_Olambda__51,axiom,
    ! [Uu2: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pa(int,fun(int,product_prod(int,int))),Uu2),Uua) = $ite(Uu2 = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu2)),Uua)),aa(int,int,abs_abs(int),Uu2))) ).

% ATP.lambda_51
tff(fact_8231_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_fz(nat,fun(nat,A),Uu2),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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_52
tff(fact_8232_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alo(fun(nat,A),fun(A,$o),Uu2),Uua)
        <=> filterlim(nat,A,Uu2,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_53
tff(fact_8233_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_sc(nat,fun(num,option(num))),Uu2),Uua) = case_nat(option(num),none(num),aTP_Lamp_sb(num,fun(nat,option(num)),Uua),Uu2) ).

% ATP.lambda_54
tff(fact_8234_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aTP_Lamp_rz(nat,fun(num,option(num)),Uu2),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_55
tff(fact_8235_ATP_Olambda__56,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ry(num,fun(nat,option(num)),Uu2),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_56
tff(fact_8236_ATP_Olambda__57,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_ue(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu2),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_57
tff(fact_8237_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_sq(list(list(A)),fun(nat,list(A)),Uu2),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sp(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu2))) ).

% ATP.lambda_58
tff(fact_8238_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_op(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_59
tff(fact_8239_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_on(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_om(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_60
tff(fact_8240_ATP_Olambda__61,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_bm(real,fun(nat,real),Uu2),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)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uu2),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_61
tff(fact_8241_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_cj(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_62
tff(fact_8242_ATP_Olambda__63,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_cp(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))),aa(nat,real,power_power(real,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_63
tff(fact_8243_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fs(nat,fun(nat,A),Uu2),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(Uu2),Uua))) ) ).

% ATP.lambda_64
tff(fact_8244_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ft(nat,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua))),Uua)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_65
tff(fact_8245_ATP_Olambda__66,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ct(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_66
tff(fact_8246_ATP_Olambda__67,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_cs(real,fun(real,$o),Uu2),Uua)
    <=> ( 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),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),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu2 ) ) ) ).

% ATP.lambda_67
tff(fact_8247_ATP_Olambda__68,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ts(code_integer,fun(code_integer,int)),Uu2),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(Uu2)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_68
tff(fact_8248_ATP_Olambda__69,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_ov(nat,fun(nat,a)),Uu2),Uua) = $let(
        m2: a,
        m2:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu2)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_69
tff(fact_8249_ATP_Olambda__70,axiom,
    ! [Uu2: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_df(complex,fun(real,$o),Uu2),Uua)
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu2) = 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_70
tff(fact_8250_ATP_Olambda__71,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_da(real,fun(int,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu2),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_71
tff(fact_8251_ATP_Olambda__72,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_db(rat,fun(int,$o),Uu2),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu2)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu2),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_72
tff(fact_8252_ATP_Olambda__73,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ap(real,fun(nat,real),Uu2),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),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu2),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_73
tff(fact_8253_ATP_Olambda__74,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_abe(real,fun(nat,real),Uu2),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_74
tff(fact_8254_ATP_Olambda__75,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aiq(fun(nat,real),fun(nat,real),Uu2),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,Uu2,Uua)) ).

% ATP.lambda_75
tff(fact_8255_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fw(nat,fun(nat,A),Uu2),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(Uu2),Uua))) ) ).

% ATP.lambda_76
tff(fact_8256_ATP_Olambda__77,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_a(nat,fun(nat,$o),Uu2),Uua)
    <=> ( aa(set(nat),$o,member(nat,Uua),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(list(nat),set(nat),set2(nat),xs)))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu2) ) ) ).

% ATP.lambda_77
tff(fact_8257_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_es(A,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_78
tff(fact_8258_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_fx(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_79
tff(fact_8259_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_fe(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_80
tff(fact_8260_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_uu(set(set(A)),fun(set(set(A)),$o),Uu2),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu2)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_81
tff(fact_8261_ATP_Olambda__82,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fj(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,power_power(nat,aa(nat,nat,binomial(Uu2),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_82
tff(fact_8262_ATP_Olambda__83,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_hp(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uu2),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_83
tff(fact_8263_ATP_Olambda__84,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_ani(nat,fun(real,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uua),Uu2)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_84
tff(fact_8264_ATP_Olambda__85,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_vi(set(A),fun(set(A),$o),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu2)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_85
tff(fact_8265_ATP_Olambda__86,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_or(set(A),fun(set(A),$o)),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu2),Uua)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_86
tff(fact_8266_ATP_Olambda__87,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uu2) ).

% ATP.lambda_87
tff(fact_8267_ATP_Olambda__88,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ep(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uua) ).

% ATP.lambda_88
tff(fact_8268_ATP_Olambda__89,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_vj(B,fun(A,set(product_prod(B,A))),Uu2),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu2),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_89
tff(fact_8269_ATP_Olambda__90,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_go(nat,fun(complex,$o),Uu2),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu2) = one_one(complex) ) ) ).

% ATP.lambda_90
tff(fact_8270_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ln(nat,fun(A,$o),Uu2),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu2) = one_one(A) ) ) ) ).

% ATP.lambda_91
tff(fact_8271_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_lf(A,fun(A,$o),Uu2),Uua)
        <=> ( aa(set(A),$o,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)),Uu2) ) ) ) ).

% ATP.lambda_92
tff(fact_8272_ATP_Olambda__93,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_akm(real,fun(nat,real),Uu2),Uua) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu2),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_93
tff(fact_8273_ATP_Olambda__94,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_ama(real,fun(real,real),Uu2),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_94
tff(fact_8274_ATP_Olambda__95,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_anj(real,fun(real,real),Uu2),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu2),Uua)),Uua) ).

% ATP.lambda_95
tff(fact_8275_ATP_Olambda__96,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ao(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu2),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_96
tff(fact_8276_ATP_Olambda__97,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_cw(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_97
tff(fact_8277_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jo(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_98
tff(fact_8278_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,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_99
tff(fact_8279_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajr(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_100
tff(fact_8280_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_101
tff(fact_8281_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_lk(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu2,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_102
tff(fact_8282_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_li(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_103
tff(fact_8283_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bg(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_104
tff(fact_8284_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_105
tff(fact_8285_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_kh(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_106
tff(fact_8286_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_107
tff(fact_8287_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ie(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_108
tff(fact_8288_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_id(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_109
tff(fact_8289_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajs(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_110
tff(fact_8290_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_em(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_111
tff(fact_8291_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aro(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y5: A] :
                ( aa(A,$o,Uu2,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y5) ) ) ) ) ).

% ATP.lambda_112
tff(fact_8292_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_arp(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y5: A] :
                ( aa(A,$o,Uu2,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Uua) ) ) ) ) ).

% ATP.lambda_113
tff(fact_8293_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jh(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu2,Uua),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_114
tff(fact_8294_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hz(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_115
tff(fact_8295_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_anx(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_116
tff(fact_8296_ATP_Olambda__117,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ado(fun(B,real),fun(B,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_117
tff(fact_8297_ATP_Olambda__118,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_gg(fun(A,real),fun(A,complex),Uu2),Uua) = complex2(aa(A,real,Uu2,Uua),zero_zero(real)) ).

% ATP.lambda_118
tff(fact_8298_ATP_Olambda__119,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_xw(fun(B,A),fun(B,set(A)),Uu2),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,Uu2,Uua)),bot_bot(set(A))) ).

% ATP.lambda_119
tff(fact_8299_ATP_Olambda__120,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_lb(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_120
tff(fact_8300_ATP_Olambda__121,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_le(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_121
tff(fact_8301_ATP_Olambda__122,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_akw(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(fun(nat,real),fun(nat,real),Uu2)),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_122
tff(fact_8302_ATP_Olambda__123,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_akv(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_aiq(fun(nat,real),fun(nat,real),Uu2)),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_123
tff(fact_8303_ATP_Olambda__124,axiom,
    ! [Uu2: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_zk(code_integer,fun($o,char)),Uu2),(Uua)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_zj($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu2)) ).

% ATP.lambda_124
tff(fact_8304_ATP_Olambda__125,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ali(fun(A,B),fun(A,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_125
tff(fact_8305_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_dj(A,fun(nat,A),Uu2),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_126
tff(fact_8306_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_du(A,fun(nat,A),Uu2),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,Uu2),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_127
tff(fact_8307_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_dm(A,fun(nat,A),Uu2),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,Uu2),Uua)) ) ).

% ATP.lambda_128
tff(fact_8308_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_di(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_129
tff(fact_8309_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_bx(A,fun(nat,A),Uu2),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,Uu2),Uua)) ) ).

% ATP.lambda_130
tff(fact_8310_ATP_Olambda__131,axiom,
    ! [Uu2: num,Uua: num] : aa(num,int,aTP_Lamp_rh(num,fun(num,int),Uu2),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu2)),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu2))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_131
tff(fact_8311_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_dt(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu2)),Uua)) ) ).

% ATP.lambda_132
tff(fact_8312_ATP_Olambda__133,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_zb(nat,fun(real,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu2)) ).

% ATP.lambda_133
tff(fact_8313_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_dn(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_134
tff(fact_8314_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_do(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_135
tff(fact_8315_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_akr(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_136
tff(fact_8316_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_akq(A,fun(nat,A),Uu2),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,Uu2),Uua)) ) ).

% ATP.lambda_137
tff(fact_8317_ATP_Olambda__138,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_gy(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_138
tff(fact_8318_ATP_Olambda__139,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_gx(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_139
tff(fact_8319_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aml(A,fun(set(A),$o),Uu2),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & aa(set(A),$o,member(A,Uu2),Uua) ) ) ) ).

% ATP.lambda_140
tff(fact_8320_ATP_Olambda__141,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_tt(code_integer,fun(code_integer,num)),Uu2),Uua) = $let(
        l2: num,
        l2:= aa(code_integer,num,code_num_of_integer,Uu2),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_141
tff(fact_8321_ATP_Olambda__142,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_tu(code_integer,fun(code_integer,nat)),Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $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_142
tff(fact_8322_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_if(nat,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu2) ) ).

% ATP.lambda_143
tff(fact_8323_ATP_Olambda__144,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_cu(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu2 ) ) ).

% ATP.lambda_144
tff(fact_8324_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_od(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),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_145
tff(fact_8325_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_oe(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_146
tff(fact_8326_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_im(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_147
tff(fact_8327_ATP_Olambda__148,axiom,
    ! [A: $tType,Uu2: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_sj(A,fun(set(set(A)),set(set(A)))),Uu2),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)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2)),Uua)) ).

% ATP.lambda_148
tff(fact_8328_ATP_Olambda__149,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fv(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu2),Uua)) ).

% ATP.lambda_149
tff(fact_8329_ATP_Olambda__150,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_dg(set(A),fun(A,$o),Uu2),Uua)
    <=> ( Uu2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_150
tff(fact_8330_ATP_Olambda__151,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ako(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_151
tff(fact_8331_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_akh(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_152
tff(fact_8332_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_uf(fun(nat,A),fun(nat,product_prod(nat,A)),Uu2),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu2,Uua)) ).

% ATP.lambda_153
tff(fact_8333_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_lz(A,fun(nat,A),Uu2),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu2),Uua))) ) ).

% ATP.lambda_154
tff(fact_8334_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_akj(fun(nat,A),fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua))) ) ).

% ATP.lambda_155
tff(fact_8335_ATP_Olambda__156,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ot(int,fun(int,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu2),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_156
tff(fact_8336_ATP_Olambda__157,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_aki(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_157
tff(fact_8337_ATP_Olambda__158,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ajy(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_158
tff(fact_8338_ATP_Olambda__159,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ye(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),Uu2),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ).

% ATP.lambda_159
tff(fact_8339_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_160
tff(fact_8340_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_161
tff(fact_8341_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_akk(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_162
tff(fact_8342_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aka(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_163
tff(fact_8343_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_akl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_164
tff(fact_8344_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ank(real,fun(A,$o),Uu2),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu2),real_V7770717601297561774m_norm(A,Uua)) ) ) ).

% ATP.lambda_165
tff(fact_8345_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ajm(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_166
tff(fact_8346_ATP_Olambda__167,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_sv(nat,fun(list(A),$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu2),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_167
tff(fact_8347_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_jr(A,fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_168
tff(fact_8348_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_md(A,fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_169
tff(fact_8349_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_jm(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_170
tff(fact_8350_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aod(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> eventually(A,Uu2,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_171
tff(fact_8351_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_te(list(A),fun(A,$o),Uu2),Uua)
    <=> aa(set(A),$o,member(A,Uua),aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_172
tff(fact_8352_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat,Uua: A] : aa(A,A,aTP_Lamp_aie(nat,fun(A,A),Uu2),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu2) ) ).

% ATP.lambda_173
tff(fact_8353_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: real] : aa(real,A,aTP_Lamp_ald(A,fun(real,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu2) ) ).

% ATP.lambda_174
tff(fact_8354_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_mi(set(A),fun(set(A),$o),Uu2),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu2) ) ).

% ATP.lambda_175
tff(fact_8355_ATP_Olambda__176,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ml(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu2) ) ).

% ATP.lambda_176
tff(fact_8356_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tw(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_177
tff(fact_8357_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_eo(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_178
tff(fact_8358_ATP_Olambda__179,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yu(nat,fun(nat,nat),Uu2),Uua) = modulo_modulo(nat,Uua,Uu2) ).

% ATP.lambda_179
tff(fact_8359_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ale(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_180
tff(fact_8360_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_qd(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_181
tff(fact_8361_ATP_Olambda__182,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ga(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu2) ) ).

% ATP.lambda_182
tff(fact_8362_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_anp(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_183
tff(fact_8363_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ha(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_184
tff(fact_8364_ATP_Olambda__185,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aji(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu2) ).

% ATP.lambda_185
tff(fact_8365_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_alb(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_186
tff(fact_8366_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_al(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_187
tff(fact_8367_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_qr(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_188
tff(fact_8368_ATP_Olambda__189,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qz(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu2) ).

% ATP.lambda_189
tff(fact_8369_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_pz(A,fun(A,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uua),Uu2) ) ).

% ATP.lambda_190
tff(fact_8370_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ql(A,fun(A,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uua),Uu2) ) ).

% ATP.lambda_191
tff(fact_8371_ATP_Olambda__192,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_aao(nat,fun(real,real),Uu2),Uua) = aa(nat,real,power_power(real,Uua),Uu2) ).

% ATP.lambda_192
tff(fact_8372_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_vf(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu2) ) ).

% ATP.lambda_193
tff(fact_8373_ATP_Olambda__194,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_xn(set(A),fun(set(A),set(A)),Uu2),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu2) ).

% ATP.lambda_194
tff(fact_8374_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_va(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu2) ) ).

% ATP.lambda_195
tff(fact_8375_ATP_Olambda__196,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_nn(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu2) ).

% ATP.lambda_196
tff(fact_8376_ATP_Olambda__197,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aTP_Lamp_qy(int,fun(int,int),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu2) ).

% ATP.lambda_197
tff(fact_8377_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_py(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2) ) ).

% ATP.lambda_198
tff(fact_8378_ATP_Olambda__199,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_aar(real,fun(real,real),Uu2),Uua) = powr(real,Uua,Uu2) ).

% ATP.lambda_199
tff(fact_8379_ATP_Olambda__200,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ld(nat,fun(nat,$o),Uu2),Uua)
    <=> dvd_dvd(nat,Uua,Uu2) ) ).

% ATP.lambda_200
tff(fact_8380_ATP_Olambda__201,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_mm(int,fun(int,$o),Uu2),Uua)
    <=> dvd_dvd(int,Uua,Uu2) ) ).

% ATP.lambda_201
tff(fact_8381_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ag(A,fun(A,$o),Uu2),Uua)
        <=> dvd_dvd(A,Uua,Uu2) ) ) ).

% ATP.lambda_202
tff(fact_8382_ATP_Olambda__203,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_uo(B,fun(A,product_prod(A,B))),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu2) ).

% ATP.lambda_203
tff(fact_8383_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu2: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ug(A,fun(nat,product_prod(nat,A)),Uu2),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu2) ).

% ATP.lambda_204
tff(fact_8384_ATP_Olambda__205,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_up(A,fun(B,product_prod(B,A))),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu2) ).

% ATP.lambda_205
tff(fact_8385_ATP_Olambda__206,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ef(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(Uua),Uu2) ).

% ATP.lambda_206
tff(fact_8386_ATP_Olambda__207,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ajw(real,fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),Uu2) ).

% ATP.lambda_207
tff(fact_8387_ATP_Olambda__208,axiom,
    ! [Uu2: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_wd(set(nat),fun(nat,$o),Uu2),Uua)
    <=> aa(set(nat),$o,member(nat,Uua),Uu2) ) ).

% ATP.lambda_208
tff(fact_8388_ATP_Olambda__209,axiom,
    ! [B: $tType,Uu2: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_arv(set(B),fun(B,$o),Uu2),Uua)
    <=> aa(set(B),$o,member(B,Uua),Uu2) ) ).

% ATP.lambda_209
tff(fact_8389_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aoa(set(A),fun(A,$o),Uu2),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu2) ) ) ).

% ATP.lambda_210
tff(fact_8390_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ua(set(A),fun(A,$o),Uu2),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu2) ) ) ).

% ATP.lambda_211
tff(fact_8391_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(A,$o)),Uu2),Uua)
    <=> aa(set(A),$o,member(A,Uua),Uu2) ) ).

% ATP.lambda_212
tff(fact_8392_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_sw(nat,fun(list(A),A),Uu2),Uua) = aa(nat,A,nth(A,Uua),Uu2) ).

% ATP.lambda_213
tff(fact_8393_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_cy(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_214
tff(fact_8394_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aqn(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_215
tff(fact_8395_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aqb(fun(A,real),fun(A,$o),Uu2),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ) ).

% ATP.lambda_216
tff(fact_8396_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aqh(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_217
tff(fact_8397_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_any(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_218
tff(fact_8398_ATP_Olambda__219,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ho(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,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_219
tff(fact_8399_ATP_Olambda__220,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_hn(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_220
tff(fact_8400_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqs(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_221
tff(fact_8401_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_ajj(fun(set(nat),A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,Uu2,aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_222
tff(fact_8402_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_ajk(fun(set(nat),A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,Uu2,aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_223
tff(fact_8403_ATP_Olambda__224,axiom,
    ! [Uu2: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_aqg(fun(real,$o),fun(real,$o),Uu2),Uua)
    <=> aa(real,$o,Uu2,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_224
tff(fact_8404_ATP_Olambda__225,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_anh(fun(real,A),fun(real,A),Uu2),Uua) = aa(real,A,Uu2,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_225
tff(fact_8405_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_aqr(fun(real,A),fun(nat,A),Uu2),Uua) = aa(real,A,Uu2,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_226
tff(fact_8406_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu2: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_are(fun(int,A),fun(nat,A),Uu2),Uua) = aa(int,A,Uu2,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_227
tff(fact_8407_ATP_Olambda__228,axiom,
    ! [Uu2: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aai(fun(real,real),fun(real,real),Uu2),Uua) = aa(real,real,Uu2,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_228
tff(fact_8408_ATP_Olambda__229,axiom,
    ! [Uu2: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_aqa(fun(real,$o),fun(real,$o),Uu2),Uua)
    <=> aa(real,$o,Uu2,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_229
tff(fact_8409_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_amb(fun(real,A),fun(real,A),Uu2),Uua) = aa(real,A,Uu2,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_230
tff(fact_8410_ATP_Olambda__231,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_rt(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_231
tff(fact_8411_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_232
tff(fact_8412_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajd(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_233
tff(fact_8413_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_234
tff(fact_8414_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_235
tff(fact_8415_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_el(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_236
tff(fact_8416_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rq(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_237
tff(fact_8417_ATP_Olambda__238,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu2: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_vp(B,fun(fun(B,A),A),Uu2),Uua) = aa(B,A,Uua,Uu2) ) ).

% ATP.lambda_238
tff(fact_8418_ATP_Olambda__239,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu2: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_vq(B,fun(fun(B,A),A),Uu2),Uua) = aa(B,A,Uua,Uu2) ) ).

% ATP.lambda_239
tff(fact_8419_ATP_Olambda__240,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu2: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_wi(A,fun(fun(A,B),B),Uu2),Uua) = aa(A,B,Uua,Uu2) ) ).

% ATP.lambda_240
tff(fact_8420_ATP_Olambda__241,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu2: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_wj(A,fun(fun(A,B),B),Uu2),Uua) = aa(A,B,Uua,Uu2) ) ).

% ATP.lambda_241
tff(fact_8421_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_sn(A,fun(option(A),option(A))),Uu2),Uua) = some(A,case_option(A,A,Uu2,aa(A,fun(A,A),ord_max(A),Uu2),Uua)) ) ).

% ATP.lambda_242
tff(fact_8422_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_tp(A,fun(option(A),option(A))),Uu2),Uua) = some(A,case_option(A,A,Uu2,aa(A,fun(A,A),sup_sup(A),Uu2),Uua)) ) ).

% ATP.lambda_243
tff(fact_8423_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_tq(A,fun(option(A),option(A))),Uu2),Uua) = some(A,case_option(A,A,Uu2,aa(A,fun(A,A),inf_inf(A),Uu2),Uua)) ) ).

% ATP.lambda_244
tff(fact_8424_ATP_Olambda__245,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aTP_Lamp_sa(nat,fun(num,option(num)),Uu2),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_245
tff(fact_8425_ATP_Olambda__246,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_rv(num,fun(nat,option(num)),Uu2),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_246
tff(fact_8426_ATP_Olambda__247,axiom,
    ! [Uu2: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_abh(fun(nat,real),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_abg(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua)) ).

% ATP.lambda_247
tff(fact_8427_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_aap(fun(nat,A),fun(A,A),Uu2),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua)) ) ).

% ATP.lambda_248
tff(fact_8428_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: real,Uua: A] : aa(A,set(A),aTP_Lamp_alz(real,fun(A,set(A)),Uu2),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aly(real,fun(A,fun(A,$o)),Uu2),Uua)) ) ).

% ATP.lambda_249
tff(fact_8429_ATP_Olambda__250,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: B] : aa(B,set(A),aTP_Lamp_xg(fun(A,fun(B,$o)),fun(B,set(A)),Uu2),Uua) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_xf(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu2),Uua)) ).

% ATP.lambda_250
tff(fact_8430_ATP_Olambda__251,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_ym(fun(C,fun(B,A)),fun(fun(B,C),A),Uu2),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_yl(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu2),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_251
tff(fact_8431_ATP_Olambda__252,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_yn(fun(C,fun(B,A)),fun(B,A),Uu2),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_yj(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu2),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_252
tff(fact_8432_ATP_Olambda__253,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_yo(fun(C,fun(B,A)),fun(fun(B,C),A),Uu2),Uua) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_yl(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu2),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_253
tff(fact_8433_ATP_Olambda__254,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_yk(fun(C,fun(B,A)),fun(B,A),Uu2),Uua) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_yj(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu2),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_254
tff(fact_8434_ATP_Olambda__255,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_hv(nat,fun(nat,complex),Uu2),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu2))) ).

% ATP.lambda_255
tff(fact_8435_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ack(A,fun(A,A),Uu2),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),Uu2)),Uua)),aa(A,A,inverse_inverse(A),Uu2))) ) ).

% ATP.lambda_256
tff(fact_8436_ATP_Olambda__257,axiom,
    ! [Uu2: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_anm(fun(real,real),fun(real,$o),Uu2),Uua)
    <=> ( aa(real,real,Uu2,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_257
tff(fact_8437_ATP_Olambda__258,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_un(fun(B,option(A)),fun(B,$o),Uu2),Uua)
    <=> ( aa(B,option(A),Uu2,Uua) != none(A) ) ) ).

% ATP.lambda_258
tff(fact_8438_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_dr(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(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,Uu2),Uua))) ) ).

% ATP.lambda_259
tff(fact_8439_ATP_Olambda__260,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_cr(nat,fun(nat,$o),Uu2),Uua)
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_260
tff(fact_8440_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ds(A,fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu2)),Uua))) ) ).

% ATP.lambda_261
tff(fact_8441_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_dp(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua))) ) ).

% ATP.lambda_262
tff(fact_8442_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_dq(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua))) ) ).

% ATP.lambda_263
tff(fact_8443_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wg(set(set(A)),fun(A,$o),Uu2),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu2)) ) ).

% ATP.lambda_264
tff(fact_8444_ATP_Olambda__265,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wf(set(set(A)),fun(A,$o),Uu2),Uua)
    <=> complete_Inf_Inf($o,aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu2)) ) ).

% ATP.lambda_265
tff(fact_8445_ATP_Olambda__266,axiom,
    ! [A: $tType,Uu2: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_yt(fun(nat,set(A)),fun(nat,set(A)),Uu2),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_266
tff(fact_8446_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_cq(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_267
tff(fact_8447_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bo(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_268
tff(fact_8448_ATP_Olambda__269,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_td(list(A),fun(A,$o),Uu2),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_269
tff(fact_8449_ATP_Olambda__270,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_xo(fun(B,set(A)),fun(set(B),set(A)),Uu2),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu2),Uua)) ).

% ATP.lambda_270
tff(fact_8450_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu2: set(A),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(set(A),fun(fun(A,$o),$o),aTP_Lamp_vr(set(A),fun(fun(A,$o),$o)),Uu2),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(A),set($o),image(A,$o,Uua),Uu2)) ) ).

% ATP.lambda_271
tff(fact_8451_ATP_Olambda__272,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_xv(fun(B,set(A)),fun(set(B),set(A)),Uu2),Uua) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),Uu2),Uua)) ).

% ATP.lambda_272
tff(fact_8452_ATP_Olambda__273,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_akg(real,fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_273
tff(fact_8453_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_aah(A,fun(A,A),Uu2),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2)) ) ).

% ATP.lambda_274
tff(fact_8454_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: A] : aa(A,filter(A),aTP_Lamp_amv(A,fun(A,filter(A)),Uu2),Uua) = aa(set(A),filter(A),principal(A),set_or5935395276787703475ssThan(A,Uu2,Uua)) ) ).

% ATP.lambda_275
tff(fact_8455_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: A] : aa(A,filter(A),aTP_Lamp_amu(A,fun(A,filter(A)),Uu2),Uua) = aa(set(A),filter(A),principal(A),set_or5935395276787703475ssThan(A,Uua,Uu2)) ) ).

% ATP.lambda_276
tff(fact_8456_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_vk(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu2),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu2),Uua)) ).

% ATP.lambda_277
tff(fact_8457_ATP_Olambda__278,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sh(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu2),Uua)) ).

% ATP.lambda_278
tff(fact_8458_ATP_Olambda__279,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sg(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu2)) ).

% ATP.lambda_279
tff(fact_8459_ATP_Olambda__280,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aj(set(A),fun(A,$o),Uu2),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),Uu2) ) ).

% ATP.lambda_280
tff(fact_8460_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aok(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_281
tff(fact_8461_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aoe(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_282
tff(fact_8462_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_anz(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_283
tff(fact_8463_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_anq(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_284
tff(fact_8464_ATP_Olambda__285,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_tc(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua != Uu2 ) ) ).

% ATP.lambda_285
tff(fact_8465_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bu(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_286
tff(fact_8466_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ex(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_287
tff(fact_8467_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bt(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_288
tff(fact_8468_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_eg(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_289
tff(fact_8469_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_jb(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_290
tff(fact_8470_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_aep(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_291
tff(fact_8471_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ja(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_292
tff(fact_8472_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_agp(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_293
tff(fact_8473_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_mu(fun(A,$o),fun(A,B),Uu2),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu2,Uua)) ) ).

% ATP.lambda_294
tff(fact_8474_ATP_Olambda__295,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aju(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_295
tff(fact_8475_ATP_Olambda__296,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_acl(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_296
tff(fact_8476_ATP_Olambda__297,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aac(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_297
tff(fact_8477_ATP_Olambda__298,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aic(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_298
tff(fact_8478_ATP_Olambda__299,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_and(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_299
tff(fact_8479_ATP_Olambda__300,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahd(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_300
tff(fact_8480_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ip(fun(B,nat),fun(B,A),Uu2),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu2,Uua)) ) ).

% ATP.lambda_301
tff(fact_8481_ATP_Olambda__302,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu2: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_eb(fun(B,nat),fun(B,A),Uu2),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu2,Uua)) ) ).

% ATP.lambda_302
tff(fact_8482_ATP_Olambda__303,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_pt(fun(A,nat),fun(A,real),Uu2),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_303
tff(fact_8483_ATP_Olambda__304,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_gh(fun(A,nat),fun(A,int),Uu2),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_304
tff(fact_8484_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_act(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_305
tff(fact_8485_ATP_Olambda__306,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajh(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_306
tff(fact_8486_ATP_Olambda__307,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ly(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_307
tff(fact_8487_ATP_Olambda__308,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_at(fun(nat,real),fun(nat,complex),Uu2),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_308
tff(fact_8488_ATP_Olambda__309,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_by(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_309
tff(fact_8489_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_be(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_310
tff(fact_8490_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ck(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_311
tff(fact_8491_ATP_Olambda__312,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_ed(fun(B,real),fun(B,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu2,Uua)) ) ).

% ATP.lambda_312
tff(fact_8492_ATP_Olambda__313,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_ir(fun(B,real),fun(B,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu2,Uua)) ) ).

% ATP.lambda_313
tff(fact_8493_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_abv(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_314
tff(fact_8494_ATP_Olambda__315,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_aej(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_315
tff(fact_8495_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_ags(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_316
tff(fact_8496_ATP_Olambda__317,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_agt(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_317
tff(fact_8497_ATP_Olambda__318,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bc(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_318
tff(fact_8498_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_319
tff(fact_8499_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_320
tff(fact_8500_ATP_Olambda__321,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_vo(fun(B,set(A)),fun(B,set(A)),Uu2),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu2,Uua)) ).

% ATP.lambda_321
tff(fact_8501_ATP_Olambda__322,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_xq(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_322
tff(fact_8502_ATP_Olambda__323,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gw(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_323
tff(fact_8503_ATP_Olambda__324,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abs(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_324
tff(fact_8504_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aab(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_325
tff(fact_8505_ATP_Olambda__326,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aib(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_326
tff(fact_8506_ATP_Olambda__327,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahk(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_327
tff(fact_8507_ATP_Olambda__328,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_anb(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_328
tff(fact_8508_ATP_Olambda__329,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahj(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_329
tff(fact_8509_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_jk(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_330
tff(fact_8510_ATP_Olambda__331,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_iq(fun(B,int),fun(B,A),Uu2),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu2,Uua)) ) ).

% ATP.lambda_331
tff(fact_8511_ATP_Olambda__332,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ec(fun(B,int),fun(B,A),Uu2),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu2,Uua)) ) ).

% ATP.lambda_332
tff(fact_8512_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_333
tff(fact_8513_ATP_Olambda__334,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_alr(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_334
tff(fact_8514_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aet(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_335
tff(fact_8515_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aja(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_336
tff(fact_8516_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_agm(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_337
tff(fact_8517_ATP_Olambda__338,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adf(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_338
tff(fact_8518_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajf(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_339
tff(fact_8519_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_agu(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ).

% ATP.lambda_340
tff(fact_8520_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abi(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcsin,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_341
tff(fact_8521_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_alq(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_342
tff(fact_8522_ATP_Olambda__343,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aev(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_343
tff(fact_8523_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adl(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arccos,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_344
tff(fact_8524_ATP_Olambda__345,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_aeb(fun(A,product_prod(B,C)),fun(A,C),Uu2),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_345
tff(fact_8525_ATP_Olambda__346,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_ahs(fun(A,product_prod(B,C)),fun(A,C),Uu2),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_346
tff(fact_8526_ATP_Olambda__347,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_aec(fun(A,product_prod(B,C)),fun(A,B),Uu2),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_347
tff(fact_8527_ATP_Olambda__348,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_ahr(fun(A,product_prod(B,C)),fun(A,B),Uu2),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_348
tff(fact_8528_ATP_Olambda__349,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aid(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_349
tff(fact_8529_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahe(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_350
tff(fact_8530_ATP_Olambda__351,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bd(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_351
tff(fact_8531_ATP_Olambda__352,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gd(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_352
tff(fact_8532_ATP_Olambda__353,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_iy(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_353
tff(fact_8533_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aiw(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_354
tff(fact_8534_ATP_Olambda__355,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afp(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_355
tff(fact_8535_ATP_Olambda__356,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aii(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,tanh(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_356
tff(fact_8536_ATP_Olambda__357,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aav(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,tanh(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_357
tff(fact_8537_ATP_Olambda__358,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahh(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,tanh(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_358
tff(fact_8538_ATP_Olambda__359,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_age(fun(A,B),fun(A,B),Uu2),Uua) = sinh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_359
tff(fact_8539_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_zu(fun(A,A),fun(A,A),Uu2),Uua) = sinh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_360
tff(fact_8540_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agf(fun(A,B),fun(A,B),Uu2),Uua) = sinh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_361
tff(fact_8541_ATP_Olambda__362,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agy(fun(A,B),fun(A,B),Uu2),Uua) = cosh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_362
tff(fact_8542_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_zt(fun(A,A),fun(A,A),Uu2),Uua) = cosh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_363
tff(fact_8543_ATP_Olambda__364,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agg(fun(A,B),fun(A,B),Uu2),Uua) = cosh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_364
tff(fact_8544_ATP_Olambda__365,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adh(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,tan(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_365
tff(fact_8545_ATP_Olambda__366,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ahg(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,tan(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_366
tff(fact_8546_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acg(fun(A,real),fun(A,real),Uu2),Uua) = sin(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_367
tff(fact_8547_ATP_Olambda__368,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agn(fun(A,B),fun(A,B),Uu2),Uua) = sin(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_368
tff(fact_8548_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aal(fun(A,A),fun(A,A),Uu2),Uua) = sin(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_369
tff(fact_8549_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ace(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,exp(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_370
tff(fact_8550_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agx(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_371
tff(fact_8551_ATP_Olambda__372,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aak(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,exp(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_372
tff(fact_8552_ATP_Olambda__373,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_agw(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_373
tff(fact_8553_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_lg(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_374
tff(fact_8554_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ahi(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,cot(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_375
tff(fact_8555_ATP_Olambda__376,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aku(fun(nat,real),fun(nat,real),Uu2),Uua) = cos(real,aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_376
tff(fact_8556_ATP_Olambda__377,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acp(fun(A,real),fun(A,real),Uu2),Uua) = cos(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_377
tff(fact_8557_ATP_Olambda__378,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ago(fun(A,B),fun(A,B),Uu2),Uua) = cos(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_378
tff(fact_8558_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aag(fun(A,A),fun(A,A),Uu2),Uua) = cos(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_379
tff(fact_8559_ATP_Olambda__380,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_mv(fun(nat,complex),fun(nat,real),Uu2),Uua) = re(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_380
tff(fact_8560_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aex(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_381
tff(fact_8561_ATP_Olambda__382,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aiu(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_382
tff(fact_8562_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ms(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_383
tff(fact_8563_ATP_Olambda__384,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_ng(fun(nat,complex),fun(nat,real),Uu2),Uua) = im(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_384
tff(fact_8564_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aey(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_385
tff(fact_8565_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aiv(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_386
tff(fact_8566_ATP_Olambda__387,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_nf(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_387
tff(fact_8567_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(nat,set(A)),Uua: nat] : aa(nat,filter(A),aTP_Lamp_arq(fun(nat,set(A)),fun(nat,filter(A)),Uu2),Uua) = aa(set(A),filter(A),principal(A),aa(nat,set(A),Uu2,Uua)) ) ).

% ATP.lambda_388
tff(fact_8568_ATP_Olambda__389,axiom,
    ! [C: $tType,D6: $tType,Uu2: fun(D6,set(C)),Uua: D6] : aa(D6,filter(C),aTP_Lamp_amo(fun(D6,set(C)),fun(D6,filter(C)),Uu2),Uua) = aa(set(C),filter(C),principal(C),aa(D6,set(C),Uu2,Uua)) ).

% ATP.lambda_389
tff(fact_8569_ATP_Olambda__390,axiom,
    ! [D6: $tType,C: $tType,Uu2: fun(C,set(D6)),Uua: C] : aa(C,filter(D6),aTP_Lamp_amn(fun(C,set(D6)),fun(C,filter(D6)),Uu2),Uua) = aa(set(D6),filter(D6),principal(D6),aa(C,set(D6),Uu2,Uua)) ).

% ATP.lambda_390
tff(fact_8570_ATP_Olambda__391,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_amh(fun(B,set(A)),fun(B,filter(A)),Uu2),Uua) = aa(set(A),filter(A),principal(A),aa(B,set(A),Uu2,Uua)) ).

% ATP.lambda_391
tff(fact_8571_ATP_Olambda__392,axiom,
    ! [E4: $tType,A: $tType,Uu2: fun(A,set(E4)),Uua: A] : aa(A,filter(E4),aTP_Lamp_amm(fun(A,set(E4)),fun(A,filter(E4)),Uu2),Uua) = aa(set(E4),filter(E4),principal(E4),aa(A,set(E4),Uu2,Uua)) ).

% ATP.lambda_392
tff(fact_8572_ATP_Olambda__393,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_amp(fun(A,set(B)),fun(A,filter(B)),Uu2),Uua) = aa(set(B),filter(B),principal(B),aa(A,set(B),Uu2,Uua)) ).

% ATP.lambda_393
tff(fact_8573_ATP_Olambda__394,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_vn(fun(A,set(B)),fun(A,nat),Uu2),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu2,Uua)) ).

% ATP.lambda_394
tff(fact_8574_ATP_Olambda__395,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_adk(fun(real,fun(nat,real)),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),Uu2,Uua)) ).

% ATP.lambda_395
tff(fact_8575_ATP_Olambda__396,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_aij(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_396
tff(fact_8576_ATP_Olambda__397,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_ev(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_397
tff(fact_8577_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_add(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_398
tff(fact_8578_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajc(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_399
tff(fact_8579_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_agv(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ).

% ATP.lambda_400
tff(fact_8580_ATP_Olambda__401,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,complex,aTP_Lamp_nj(fun(nat,complex),fun(nat,complex),Uu2),Uua) = cnj(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_401
tff(fact_8581_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aez(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_402
tff(fact_8582_ATP_Olambda__403,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aiz(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_403
tff(fact_8583_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_ni(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_404
tff(fact_8584_ATP_Olambda__405,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_sk(fun(A,B),fun(A,fun(set(B),set(B))),Uu2),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_405
tff(fact_8585_ATP_Olambda__406,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_xm(fun(B,set(A)),fun(B,set(set(A))),Uu2),Uua) = pow2(A,aa(B,set(A),Uu2,Uua)) ).

% ATP.lambda_406
tff(fact_8586_ATP_Olambda__407,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_nc(fun(A,nat),fun(A,nat),Uu2),Uua) = aa(nat,nat,suc,aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_407
tff(fact_8587_ATP_Olambda__408,axiom,
    ! [B: $tType,Uu2: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_qb(fun(B,$o),fun(B,$o),Uu2),Uua)
    <=> ~ aa(B,$o,Uu2,Uua) ) ).

% ATP.lambda_408
tff(fact_8588_ATP_Olambda__409,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ady(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ~ aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_409
tff(fact_8589_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ai(fun(A,$o),fun(A,$o),Uu2),Uua)
    <=> ~ aa(A,$o,Uu2,Uua) ) ).

% ATP.lambda_410
tff(fact_8590_ATP_Olambda__411,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu2: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_arl(fun(B,fun(A,$o)),fun(B,$o),Uu2),Uua)
        <=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu2,Uua),X_12) ) ) ).

% ATP.lambda_411
tff(fact_8591_ATP_Olambda__412,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_arm(fun(A,fun(B,$o)),fun(A,$o),Uu2),Uua)
    <=> ! [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu2,Uua),X_12) ) ).

% ATP.lambda_412
tff(fact_8592_ATP_Olambda__413,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: A,Uua: real] : aa(real,filter(A),aTP_Lamp_amt(A,fun(real,filter(A)),Uu2),Uua) = aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_ams(A,fun(real,fun(A,$o)),Uu2),Uua))) ) ).

% ATP.lambda_413
tff(fact_8593_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu2: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_aim(fun(real,A),fun(real,real),Uu2),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu2,Uua))) ) ).

% ATP.lambda_414
tff(fact_8594_ATP_Olambda__415,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_aha(fun(A,B),fun(A,C),Uu2),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu2,Uua))) ) ).

% ATP.lambda_415
tff(fact_8595_ATP_Olambda__416,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_agz(fun(A,B),fun(A,C),Uu2),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu2,Uua))) ) ).

% ATP.lambda_416
tff(fact_8596_ATP_Olambda__417,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wk(set(set(A)),fun(A,$o),Uu2),Uua)
    <=> ? [X2: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X2),Uu2)
          & aa(set(A),$o,member(A,Uua),X2) ) ) ).

% ATP.lambda_417
tff(fact_8597_ATP_Olambda__418,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_arj(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y5)
             => aa(A,$o,Uu2,Y5) ) ) ) ).

% ATP.lambda_418
tff(fact_8598_ATP_Olambda__419,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cg(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_419
tff(fact_8599_ATP_Olambda__420,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hm(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uub),aa(nat,real,Uu2,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_420
tff(fact_8600_ATP_Olambda__421,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_tr(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu2)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu2))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_421
tff(fact_8601_ATP_Olambda__422,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_of(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu2),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu2)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu2))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_422
tff(fact_8602_ATP_Olambda__423,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_og(num,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu2)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu2))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_423
tff(fact_8603_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_oh(num,fun(A,fun(A,product_prod(A,A))),Uu2),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu2)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu2))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_424
tff(fact_8604_ATP_Olambda__425,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ke(set(nat),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(aa(set(nat),$o,member(nat,Uub),Uu2),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_425
tff(fact_8605_ATP_Olambda__426,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_la(fun(A,B),fun(set(A),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu2,Uub),zero_zero(B)) ) ).

% ATP.lambda_426
tff(fact_8606_ATP_Olambda__427,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_lc(fun(A,B),fun(set(A),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_427
tff(fact_8607_ATP_Olambda__428,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ka(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_428
tff(fact_8608_ATP_Olambda__429,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kc(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_429
tff(fact_8609_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_au(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_430
tff(fact_8610_ATP_Olambda__431,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jz(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_431
tff(fact_8611_ATP_Olambda__432,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kb(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_432
tff(fact_8612_ATP_Olambda__433,axiom,
    ! [Uu2: 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_ty(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu2)),Uub))) ).

% ATP.lambda_433
tff(fact_8613_ATP_Olambda__434,axiom,
    ! [Uu2: 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_uh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),Uu2)),Uub))) ).

% ATP.lambda_434
tff(fact_8614_ATP_Olambda__435,axiom,
    ! [Uu2: 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_tx(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uu2),Uub))) ).

% ATP.lambda_435
tff(fact_8615_ATP_Olambda__436,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_tn(fun(A,$o),fun(A,fun(set(A),set(A))),Uu2),Uua),Uub) = $ite(aa(A,$o,Uu2,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_436
tff(fact_8616_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_kd(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(aa(nat,$o,Uu2,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_437
tff(fact_8617_ATP_Olambda__438,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_tb(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu2),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu2,Uub),zero_zero(A)) ) ).

% ATP.lambda_438
tff(fact_8618_ATP_Olambda__439,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_kx(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),zero_zero(B)) ) ).

% ATP.lambda_439
tff(fact_8619_ATP_Olambda__440,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ky(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_440
tff(fact_8620_ATP_Olambda__441,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_tk(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu2),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),some(A,aa(B,A,Uu2,Uub)),none(A)) ).

% ATP.lambda_441
tff(fact_8621_ATP_Olambda__442,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_ann(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu2),Uua),Uub)
    <=> has_field_derivative(real,Uu2,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_442
tff(fact_8622_ATP_Olambda__443,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_aln(filter(B),fun(fun(B,A),fun(A,$o)),Uu2),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu2) ) ) ).

% ATP.lambda_443
tff(fact_8623_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: $o] :
      aa($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_yy(set(A),fun(set(A),fun($o,set(A))),Uu2),Uua),(Uub)) = $ite((Uub),Uu2,Uua) ).

% ATP.lambda_444
tff(fact_8624_ATP_Olambda__445,axiom,
    ! [D6: $tType,C: $tType,B: $tType,A: $tType,Uu2: fun(D6,fun(B,C)),Uua: fun(A,D6),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D6),fun(product_prod(A,B),C),aTP_Lamp_ub(fun(D6,fun(B,C)),fun(fun(A,D6),fun(product_prod(A,B),C)),Uu2),Uua),Uub) = aa(B,C,aa(D6,fun(B,C),Uu2,aa(A,D6,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_445
tff(fact_8625_ATP_Olambda__446,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_yl(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu2),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu2,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_446
tff(fact_8626_ATP_Olambda__447,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_447
tff(fact_8627_ATP_Olambda__448,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_om(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_448
tff(fact_8628_ATP_Olambda__449,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_adj(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu2,Uub),Uua) ).

% ATP.lambda_449
tff(fact_8629_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_450
tff(fact_8630_ATP_Olambda__451,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_451
tff(fact_8631_ATP_Olambda__452,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_yj(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu2),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_452
tff(fact_8632_ATP_Olambda__453,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_vs(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu2),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_453
tff(fact_8633_ATP_Olambda__454,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu2: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_vu(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu2),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_454
tff(fact_8634_ATP_Olambda__455,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_aae(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu2),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu2,Uub),Uua) ) ).

% ATP.lambda_455
tff(fact_8635_ATP_Olambda__456,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_wm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_456
tff(fact_8636_ATP_Olambda__457,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_iv(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_457
tff(fact_8637_ATP_Olambda__458,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_gm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_458
tff(fact_8638_ATP_Olambda__459,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu2: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ark(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu2,Uub),Uua) ) ) ).

% ATP.lambda_459
tff(fact_8639_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_rn(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_460
tff(fact_8640_ATP_Olambda__461,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_afk(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_461
tff(fact_8641_ATP_Olambda__462,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aht(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_462
tff(fact_8642_ATP_Olambda__463,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_st(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu2,Uub),Uua) ).

% ATP.lambda_463
tff(fact_8643_ATP_Olambda__464,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ej(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu2,Uub),Uua) ) ).

% ATP.lambda_464
tff(fact_8644_ATP_Olambda__465,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_abq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_465
tff(fact_8645_ATP_Olambda__466,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_acx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_466
tff(fact_8646_ATP_Olambda__467,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_afe(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_467
tff(fact_8647_ATP_Olambda__468,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_afc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_468
tff(fact_8648_ATP_Olambda__469,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ahv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_469
tff(fact_8649_ATP_Olambda__470,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xf(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_470
tff(fact_8650_ATP_Olambda__471,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_afi(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_471
tff(fact_8651_ATP_Olambda__472,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_afg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_472
tff(fact_8652_ATP_Olambda__473,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ahx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_473
tff(fact_8653_ATP_Olambda__474,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_kr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_474
tff(fact_8654_ATP_Olambda__475,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_kn(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_475
tff(fact_8655_ATP_Olambda__476,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_su(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu2),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu2,Uub),Uua) ).

% ATP.lambda_476
tff(fact_8656_ATP_Olambda__477,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ss(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_477
tff(fact_8657_ATP_Olambda__478,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gc(A,fun(A,fun(nat,A)),Uu2),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_gb(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_478
tff(fact_8658_ATP_Olambda__479,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_ey(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_479
tff(fact_8659_ATP_Olambda__480,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ea(A,fun(A,fun(nat,A)),Uu2),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_dz(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_480
tff(fact_8660_ATP_Olambda__481,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dy(A,fun(A,fun(nat,A)),Uu2),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_dx(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_481
tff(fact_8661_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cd(nat,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu2,one_one(A),zero_zero(A))),
            aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_482
tff(fact_8662_ATP_Olambda__483,axiom,
    ! [Uu2: 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_tv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu2),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($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)),Uu2),Uua,aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_483
tff(fact_8663_ATP_Olambda__484,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pd(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu2,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_484
tff(fact_8664_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_485
tff(fact_8665_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),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_ia(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_486
tff(fact_8666_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ail(fun(nat,A),fun(nat,fun(A,A)),Uu2),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_bi(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_487
tff(fact_8667_ATP_Olambda__488,axiom,
    ! [Uu2: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_zj($o,fun(code_integer,fun($o,char)),(Uu2)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_zi($o,fun($o,fun(code_integer,fun($o,char))),(Uu2)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_488
tff(fact_8668_ATP_Olambda__489,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ps(rat,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_489
tff(fact_8669_ATP_Olambda__490,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pq(rat,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_490
tff(fact_8670_ATP_Olambda__491,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pn(rat,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_491
tff(fact_8671_ATP_Olambda__492,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pl(rat,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_492
tff(fact_8672_ATP_Olambda__493,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pj(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pi(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_493
tff(fact_8673_ATP_Olambda__494,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pg(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pf(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_494
tff(fact_8674_ATP_Olambda__495,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afj(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_afi(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_495
tff(fact_8675_ATP_Olambda__496,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afh(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_afg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_496
tff(fact_8676_ATP_Olambda__497,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aff(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_afe(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_497
tff(fact_8677_ATP_Olambda__498,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_afd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_afc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_498
tff(fact_8678_ATP_Olambda__499,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_acy(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_acx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_499
tff(fact_8679_ATP_Olambda__500,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_iw(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_iv(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub),Uua) ) ).

% ATP.lambda_500
tff(fact_8680_ATP_Olambda__501,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ahy(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),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_ahx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_501
tff(fact_8681_ATP_Olambda__502,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ahw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),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_ahv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_502
tff(fact_8682_ATP_Olambda__503,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),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_abq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_503
tff(fact_8683_ATP_Olambda__504,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_gn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_gm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub)),Uua) ) ).

% ATP.lambda_504
tff(fact_8684_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ek(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu2),Uua),Uub) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(nat,fun(A,B),aTP_Lamp_ej(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_505
tff(fact_8685_ATP_Olambda__506,axiom,
    ! [D6: $tType,E4: $tType,A: $tType,C: $tType,B: $tType,Uu2: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D6,E4)] : aa(product_prod(D6,E4),set(A),aa(product_prod(B,C),fun(product_prod(D6,E4),set(A)),aTP_Lamp_yc(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(B,C),fun(product_prod(D6,E4),set(A))),Uu2),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D6,E4),fun(B,fun(C,set(A))),aTP_Lamp_yb(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(D6,E4),fun(B,fun(C,set(A)))),Uu2),Uub)),Uua) ).

% ATP.lambda_506
tff(fact_8686_ATP_Olambda__507,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aaw(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_507
tff(fact_8687_ATP_Olambda__508,axiom,
    ! [Uu2: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aax(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_508
tff(fact_8688_ATP_Olambda__509,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu2: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_hk(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ) ).

% ATP.lambda_509
tff(fact_8689_ATP_Olambda__510,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ma(fun(nat,A),fun(nat,fun(A,$o)),Uu2),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_en(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_510
tff(fact_8690_ATP_Olambda__511,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aif(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_511
tff(fact_8691_ATP_Olambda__512,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aed(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_512
tff(fact_8692_ATP_Olambda__513,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ki(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8693_ATP_Olambda__514,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aea(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_514
tff(fact_8694_ATP_Olambda__515,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aee(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_515
tff(fact_8695_ATP_Olambda__516,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_abf(fun(nat,real),fun(real,fun(nat,real)),Uu2),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,Uu2,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_516
tff(fact_8696_ATP_Olambda__517,axiom,
    ! [Uu2: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hl(real,fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_517
tff(fact_8697_ATP_Olambda__518,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_nw(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,minus_minus(nat,Uu2),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_518
tff(fact_8698_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(A,fun(nat,A)),Uu2),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,Uu2,Uub))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_519
tff(fact_8699_ATP_Olambda__520,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_nx(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_520
tff(fact_8700_ATP_Olambda__521,axiom,
    ! [A: $tType,Uu2: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_nm(nat,fun(set(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & 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_521
tff(fact_8701_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_nl(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2) ) ) ).

% ATP.lambda_522
tff(fact_8702_ATP_Olambda__523,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_nr(nat,fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & 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_523
tff(fact_8703_ATP_Olambda__524,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_nh(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_524
tff(fact_8704_ATP_Olambda__525,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_nk(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_525
tff(fact_8705_ATP_Olambda__526,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_nv(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_526
tff(fact_8706_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aas(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu2)),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_527
tff(fact_8707_ATP_Olambda__528,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_na(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uub)),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_528
tff(fact_8708_ATP_Olambda__529,axiom,
    ! [Uu2: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_np(nat,fun(nat,fun(set(nat),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(set(nat)),$o,member(set(nat),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu2)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_529
tff(fact_8709_ATP_Olambda__530,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rs(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),vEBT_set_vebt(Uu2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_530
tff(fact_8710_ATP_Olambda__531,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rl(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),vEBT_set_vebt(Uu2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_531
tff(fact_8711_ATP_Olambda__532,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_tf(fun(A,$o),fun(list(A),fun(nat,$o)),Uu2),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,Uu2,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_532
tff(fact_8712_ATP_Olambda__533,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_px(list(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),Uu2))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_533
tff(fact_8713_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_sy(fun(A,$o),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_534
tff(fact_8714_ATP_Olambda__535,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mc(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_535
tff(fact_8715_ATP_Olambda__536,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu2: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_tz(set(A),fun(nat,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,Uu2,Uua)),Uub) ) ) ) ).

% ATP.lambda_536
tff(fact_8716_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ru(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,Uua),Uu2)) ) ) ) ) ).

% ATP.lambda_537
tff(fact_8717_ATP_Olambda__538,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_mw(set(A),fun(nat,fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu2)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_538
tff(fact_8718_ATP_Olambda__539,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_mz(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_539
tff(fact_8719_ATP_Olambda__540,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ch(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_540
tff(fact_8720_ATP_Olambda__541,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cn(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_541
tff(fact_8721_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(A,fun(fun(nat,A),fun(nat,A)),Uu2),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,Uu2),Uub)) ) ).

% ATP.lambda_542
tff(fact_8722_ATP_Olambda__543,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_et(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu2),Uub)) ).

% ATP.lambda_543
tff(fact_8723_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jt(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu2)
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_544
tff(fact_8724_ATP_Olambda__545,axiom,
    ! [A: $tType,Uu2: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_af(A,fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_545
tff(fact_8725_ATP_Olambda__546,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_oz(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_546
tff(fact_8726_ATP_Olambda__547,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_mg(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_547
tff(fact_8727_ATP_Olambda__548,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ow(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_548
tff(fact_8728_ATP_Olambda__549,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_mj(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_549
tff(fact_8729_ATP_Olambda__550,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_mk(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_550
tff(fact_8730_ATP_Olambda__551,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_mh(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_551
tff(fact_8731_ATP_Olambda__552,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rf(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu2),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_552
tff(fact_8732_ATP_Olambda__553,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_re(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu2),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_553
tff(fact_8733_ATP_Olambda__554,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_um(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( dvd_dvd(nat,Uub,Uua)
        & dvd_dvd(nat,Uub,Uu2) ) ) ).

% ATP.lambda_554
tff(fact_8734_ATP_Olambda__555,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ui(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( dvd_dvd(int,Uub,Uua)
        & dvd_dvd(int,Uub,Uu2) ) ) ).

% ATP.lambda_555
tff(fact_8735_ATP_Olambda__556,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jw(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu2)
        & aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_556
tff(fact_8736_ATP_Olambda__557,axiom,
    ! [A: $tType,Uu2: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ns(list(A),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu2),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_557
tff(fact_8737_ATP_Olambda__558,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_ny(fun(A,nat),fun(list(A),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_558
tff(fact_8738_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_apf(fun(A,$o),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
       => aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_559
tff(fact_8739_ATP_Olambda__560,axiom,
    ! [A: $tType,Uu2: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(set(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu2)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_560
tff(fact_8740_ATP_Olambda__561,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_to(fun(A,$o),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_561
tff(fact_8741_ATP_Olambda__562,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dd(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_562
tff(fact_8742_ATP_Olambda__563,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yz(fun(A,$o),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_563
tff(fact_8743_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_de(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_564
tff(fact_8744_ATP_Olambda__565,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_565
tff(fact_8745_ATP_Olambda__566,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_566
tff(fact_8746_ATP_Olambda__567,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_kv(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_567
tff(fact_8747_ATP_Olambda__568,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_mo(fun(B,A),fun(set(B),fun(B,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu2,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_568
tff(fact_8748_ATP_Olambda__569,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu2)
            & ~ dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2)),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_569
tff(fact_8749_ATP_Olambda__570,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ak(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu2)
        & ~ aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_570
tff(fact_8750_ATP_Olambda__571,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pw(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu2) ) ).

% ATP.lambda_571
tff(fact_8751_ATP_Olambda__572,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ol(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_572
tff(fact_8752_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_alt(A,fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_573
tff(fact_8753_ATP_Olambda__574,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aly(real,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu2) ) ) ).

% ATP.lambda_574
tff(fact_8754_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ams(A,fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu2)),Uua) ) ) ).

% ATP.lambda_575
tff(fact_8755_ATP_Olambda__576,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_oq(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_576
tff(fact_8756_ATP_Olambda__577,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qv(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu2)),Uua) ) ).

% ATP.lambda_577
tff(fact_8757_ATP_Olambda__578,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qt(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_578
tff(fact_8758_ATP_Olambda__579,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qu(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu2)),Uua) ) ).

% ATP.lambda_579
tff(fact_8759_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qs(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8760_ATP_Olambda__581,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_oa(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu2),Uua),Uub)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu2) ) ).

% ATP.lambda_581
tff(fact_8761_ATP_Olambda__582,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aru(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu2) ) ).

% ATP.lambda_582
tff(fact_8762_ATP_Olambda__583,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sp(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu2),Uub)),Uua) ).

% ATP.lambda_583
tff(fact_8763_ATP_Olambda__584,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_gk(nat,fun(complex,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu2) = Uua ) ) ).

% ATP.lambda_584
tff(fact_8764_ATP_Olambda__585,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_js(complex,fun(nat,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_585
tff(fact_8765_ATP_Olambda__586,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kz(A,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_586
tff(fact_8766_ATP_Olambda__587,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_587
tff(fact_8767_ATP_Olambda__588,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_588
tff(fact_8768_ATP_Olambda__589,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_akt(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu2,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_589
tff(fact_8769_ATP_Olambda__590,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_abg(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_590
tff(fact_8770_ATP_Olambda__591,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_as(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_591
tff(fact_8771_ATP_Olambda__592,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fg(fun(nat,nat),fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,power_power(nat,Uua),Uub)) ).

% ATP.lambda_592
tff(fact_8772_ATP_Olambda__593,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_ain(fun(nat,B),fun(B,fun(nat,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu2,Uub)),aa(nat,B,power_power(B,Uua),Uub)) ) ).

% ATP.lambda_593
tff(fact_8773_ATP_Olambda__594,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_594
tff(fact_8774_ATP_Olambda__595,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_595
tff(fact_8775_ATP_Olambda__596,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_596
tff(fact_8776_ATP_Olambda__597,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cm(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_597
tff(fact_8777_ATP_Olambda__598,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_er(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_598
tff(fact_8778_ATP_Olambda__599,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_599
tff(fact_8779_ATP_Olambda__600,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kj(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,Uu2,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_600
tff(fact_8780_ATP_Olambda__601,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abn(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_601
tff(fact_8781_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeq(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_602
tff(fact_8782_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agh(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_603
tff(fact_8783_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_alu(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_604
tff(fact_8784_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_alv(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_605
tff(fact_8785_ATP_Olambda__606,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_oc(fun(B,A),fun(B,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu2,Uua)),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_606
tff(fact_8786_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_aom(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_607
tff(fact_8787_ATP_Olambda__608,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoz(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_608
tff(fact_8788_ATP_Olambda__609,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ant(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_609
tff(fact_8789_ATP_Olambda__610,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_apu(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_610
tff(fact_8790_ATP_Olambda__611,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqq(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu2,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_611
tff(fact_8791_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_iz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_612
tff(fact_8792_ATP_Olambda__613,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_613
tff(fact_8793_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aci(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8794_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aad(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_615
tff(fact_8795_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_616
tff(fact_8796_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_anc(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_617
tff(fact_8797_ATP_Olambda__618,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_618
tff(fact_8798_ATP_Olambda__619,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_619
tff(fact_8799_ATP_Olambda__620,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ano(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu2,Uub)) ).

% ATP.lambda_620
tff(fact_8800_ATP_Olambda__621,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_mf(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_621
tff(fact_8801_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ard(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_622
tff(fact_8802_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_me(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8803_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_it(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_624
tff(fact_8804_ATP_Olambda__625,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_625
tff(fact_8805_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_zq(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_626
tff(fact_8806_ATP_Olambda__627,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_627
tff(fact_8807_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_628
tff(fact_8808_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_amq(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_629
tff(fact_8809_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_630
tff(fact_8810_ATP_Olambda__631,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_631
tff(fact_8811_ATP_Olambda__632,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_632
tff(fact_8812_ATP_Olambda__633,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_633
tff(fact_8813_ATP_Olambda__634,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ane(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_634
tff(fact_8814_ATP_Olambda__635,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ge(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_635
tff(fact_8815_ATP_Olambda__636,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ajt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu2,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_636
tff(fact_8816_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_637
tff(fact_8817_ATP_Olambda__638,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gu(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_638
tff(fact_8818_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aby(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_639
tff(fact_8819_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_zx(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_640
tff(fact_8820_ATP_Olambda__641,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aia(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_641
tff(fact_8821_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_642
tff(fact_8822_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_643
tff(fact_8823_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_644
tff(fact_8824_ATP_Olambda__645,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_645
tff(fact_8825_ATP_Olambda__646,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hc(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_646
tff(fact_8826_ATP_Olambda__647,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_gi(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_647
tff(fact_8827_ATP_Olambda__648,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_648
tff(fact_8828_ATP_Olambda__649,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_agj(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_649
tff(fact_8829_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_agi(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_650
tff(fact_8830_ATP_Olambda__651,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_651
tff(fact_8831_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_652
tff(fact_8832_ATP_Olambda__653,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_agq(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu2),Uua),Uub) = complex2(aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_653
tff(fact_8833_ATP_Olambda__654,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu2,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_654
tff(fact_8834_ATP_Olambda__655,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ws(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_655
tff(fact_8835_ATP_Olambda__656,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xi(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu2,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_656
tff(fact_8836_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wt(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_657
tff(fact_8837_ATP_Olambda__658,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_658
tff(fact_8838_ATP_Olambda__659,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gt(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_659
tff(fact_8839_ATP_Olambda__660,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_660
tff(fact_8840_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_zw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_661
tff(fact_8841_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afy(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_662
tff(fact_8842_ATP_Olambda__663,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ana(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_663
tff(fact_8843_ATP_Olambda__664,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_apz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_664
tff(fact_8844_ATP_Olambda__665,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aga(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_665
tff(fact_8845_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_666
tff(fact_8846_ATP_Olambda__667,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aau(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_667
tff(fact_8847_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_668
tff(fact_8848_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajg(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_669
tff(fact_8849_ATP_Olambda__670,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_670
tff(fact_8850_ATP_Olambda__671,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_akb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_671
tff(fact_8851_ATP_Olambda__672,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeu(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_672
tff(fact_8852_ATP_Olambda__673,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aes(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_673
tff(fact_8853_ATP_Olambda__674,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_afo(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_674
tff(fact_8854_ATP_Olambda__675,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_675
tff(fact_8855_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_all(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,$o,Uu2,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_676
tff(fact_8856_ATP_Olambda__677,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ju(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_677
tff(fact_8857_ATP_Olambda__678,axiom,
    ! [B: $tType,Uu2: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_us(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu2),Uua),Uub)
    <=> ( aa(B,$o,Uu2,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_678
tff(fact_8858_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,$o,Uu2,Uub)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_679
tff(fact_8859_ATP_Olambda__680,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_jx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_680
tff(fact_8860_ATP_Olambda__681,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_681
tff(fact_8861_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aqw(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ( aa(nat,A,Uu2,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_682
tff(fact_8862_ATP_Olambda__683,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_aor(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> ( aa(B,A,Uu2,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_683
tff(fact_8863_ATP_Olambda__684,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoc(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_684
tff(fact_8864_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_aof(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,A,Uu2,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_685
tff(fact_8865_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_apq(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_686
tff(fact_8866_ATP_Olambda__687,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aog(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_687
tff(fact_8867_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aow(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_688
tff(fact_8868_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aoy(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_689
tff(fact_8869_ATP_Olambda__690,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_kg(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_690
tff(fact_8870_ATP_Olambda__691,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aqp(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_691
tff(fact_8871_ATP_Olambda__692,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_zo(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu2,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_692
tff(fact_8872_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,real),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_693
tff(fact_8873_ATP_Olambda__694,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ee(fun(B,real),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_694
tff(fact_8874_ATP_Olambda__695,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_abx(fun(A,real),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_695
tff(fact_8875_ATP_Olambda__696,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_iu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_696
tff(fact_8876_ATP_Olambda__697,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_afl(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu2) ) ).

% ATP.lambda_697
tff(fact_8877_ATP_Olambda__698,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_gl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uu2,Uub)),Uua) ) ).

% ATP.lambda_698
tff(fact_8878_ATP_Olambda__699,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_aaf(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu2),Uua),Uub) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),Uua,Uub)),Uu2) ) ).

% ATP.lambda_699
tff(fact_8879_ATP_Olambda__700,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ahu(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),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)),Uu2) ) ).

% ATP.lambda_700
tff(fact_8880_ATP_Olambda__701,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_alw(fun(A,B),fun(B,fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),Uua) ) ).

% ATP.lambda_701
tff(fact_8881_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_aqy(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_702
tff(fact_8882_ATP_Olambda__703,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kk(fun(nat,nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu2,Uub)),Uua) ) ).

% ATP.lambda_703
tff(fact_8883_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_uw(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ) ).

% ATP.lambda_704
tff(fact_8884_ATP_Olambda__705,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aps(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_705
tff(fact_8885_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apr(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_706
tff(fact_8886_ATP_Olambda__707,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apk(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_707
tff(fact_8887_ATP_Olambda__708,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anr(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_708
tff(fact_8888_ATP_Olambda__709,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_eh(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_709
tff(fact_8889_ATP_Olambda__710,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_710
tff(fact_8890_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gv(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_711
tff(fact_8891_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zz(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_712
tff(fact_8892_ATP_Olambda__713,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ls(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_713
tff(fact_8893_ATP_Olambda__714,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahc(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_714
tff(fact_8894_ATP_Olambda__715,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ait(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_715
tff(fact_8895_ATP_Olambda__716,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_apd(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_716
tff(fact_8896_ATP_Olambda__717,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_717
tff(fact_8897_ATP_Olambda__718,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anv(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_718
tff(fact_8898_ATP_Olambda__719,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_719
tff(fact_8899_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gr(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_720
tff(fact_8900_ATP_Olambda__721,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acb(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_721
tff(fact_8901_ATP_Olambda__722,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zs(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_722
tff(fact_8902_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_afu(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_723
tff(fact_8903_ATP_Olambda__724,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahp(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_724
tff(fact_8904_ATP_Olambda__725,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_afv(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_725
tff(fact_8905_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aco(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu2) ) ).

% ATP.lambda_726
tff(fact_8906_ATP_Olambda__727,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ca(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_727
tff(fact_8907_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_air(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_728
tff(fact_8908_ATP_Olambda__729,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_adu(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_729
tff(fact_8909_ATP_Olambda__730,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_he(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_730
tff(fact_8910_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_xd(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_731
tff(fact_8911_ATP_Olambda__732,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahm(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_732
tff(fact_8912_ATP_Olambda__733,axiom,
    ! [Uu2: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_aaq(fun(real,real),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uu2,Uub)),Uua) ).

% ATP.lambda_733
tff(fact_8913_ATP_Olambda__734,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_ix(fun(B,A),fun(nat,fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_734
tff(fact_8914_ATP_Olambda__735,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_acr(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_735
tff(fact_8915_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_736
tff(fact_8916_ATP_Olambda__737,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_agl(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_737
tff(fact_8917_ATP_Olambda__738,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aqk(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_738
tff(fact_8918_ATP_Olambda__739,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aew(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_739
tff(fact_8919_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_agk(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_740
tff(fact_8920_ATP_Olambda__741,axiom,
    ! [Uu2: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_amz(nat,fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu2) ).

% ATP.lambda_741
tff(fact_8921_ATP_Olambda__742,axiom,
    ! [A: $tType,Uu2: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(nat,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu2) ).

% ATP.lambda_742
tff(fact_8922_ATP_Olambda__743,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wc(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_743
tff(fact_8923_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vg(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_744
tff(fact_8924_ATP_Olambda__745,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_wz(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_745
tff(fact_8925_ATP_Olambda__746,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vb(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_746
tff(fact_8926_ATP_Olambda__747,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xs(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_747
tff(fact_8927_ATP_Olambda__748,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_arc(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_748
tff(fact_8928_ATP_Olambda__749,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_749
tff(fact_8929_ATP_Olambda__750,axiom,
    ! [Uu2: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_aat(fun(real,real),fun(real,fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),Uua) ).

% ATP.lambda_750
tff(fact_8930_ATP_Olambda__751,axiom,
    ! [A: $tType,Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_anf(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu2) ).

% ATP.lambda_751
tff(fact_8931_ATP_Olambda__752,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_aph(fun(B,A),fun(set(A),fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(set(A),$o,member(A,aa(B,A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_752
tff(fact_8932_ATP_Olambda__753,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_apg(fun(A,B),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(set(B),$o,member(B,aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_753
tff(fact_8933_ATP_Olambda__754,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aoq(fun(A,B),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(set(B),$o,member(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_754
tff(fact_8934_ATP_Olambda__755,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_aoh(fun(A,$o),fun($o,fun(A,$o)),Uu2),(Uua)),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        | (Uua) ) ) ).

% ATP.lambda_755
tff(fact_8935_ATP_Olambda__756,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aot(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = Uua ) ) ) ).

% ATP.lambda_756
tff(fact_8936_ATP_Olambda__757,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aov(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = Uua ) ) ) ).

% ATP.lambda_757
tff(fact_8937_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_758
tff(fact_8938_ATP_Olambda__759,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mb(nat,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu2),Uub))) ) ).

% ATP.lambda_759
tff(fact_8939_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bw(real,fun(fun(nat,A),fun(nat,real)),Uu2),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,Uu2),Uub)) ) ).

% ATP.lambda_760
tff(fact_8940_ATP_Olambda__761,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_arf(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_761
tff(fact_8941_ATP_Olambda__762,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kf(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu2,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_762
tff(fact_8942_ATP_Olambda__763,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_aqx(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_763
tff(fact_8943_ATP_Olambda__764,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apw(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu2,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_764
tff(fact_8944_ATP_Olambda__765,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apx(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu2,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_765
tff(fact_8945_ATP_Olambda__766,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aqu(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))),Uua) ) ) ).

% ATP.lambda_766
tff(fact_8946_ATP_Olambda__767,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hs(A,fun(nat,fun(nat,A)),Uu2),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_767
tff(fact_8947_ATP_Olambda__768,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aqo(fun(A,B),fun(B,fun(A,$o)),Uu2),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,Uu2,Uub)) ) ) ).

% ATP.lambda_768
tff(fact_8948_ATP_Olambda__769,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_art(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu2),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] :
            ( aa(set(A),$o,member(A,X2),Uua)
           => ? [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),Uub)
                & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2)),Uu2) ) ) ) ) ).

% ATP.lambda_769
tff(fact_8949_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lm(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu2,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_770
tff(fact_8950_ATP_Olambda__771,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lx(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),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_771
tff(fact_8951_ATP_Olambda__772,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ll(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),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_772
tff(fact_8952_ATP_Olambda__773,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ij(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_773
tff(fact_8953_ATP_Olambda__774,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ih(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_774
tff(fact_8954_ATP_Olambda__775,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hj(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_775
tff(fact_8955_ATP_Olambda__776,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_akf(real,fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_776
tff(fact_8956_ATP_Olambda__777,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fl(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_777
tff(fact_8957_ATP_Olambda__778,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ii(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_778
tff(fact_8958_ATP_Olambda__779,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_tl(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu2),Uua),Uub) = aa(list(A),list(A),cons(A,Uub),map_filter(B,A,Uu2,Uua)) ).

% ATP.lambda_779
tff(fact_8959_ATP_Olambda__780,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_aqc(real,fun(real,fun(real,$o)),Uu2),Uua),Uub)
    <=> aa(set(real),$o,member(real,Uub),set_or5935395276787703475ssThan(real,Uu2,Uua)) ) ).

% ATP.lambda_780
tff(fact_8960_ATP_Olambda__781,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ei(real,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_781
tff(fact_8961_ATP_Olambda__782,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(real,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_782
tff(fact_8962_ATP_Olambda__783,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_abw(fun(A,B),fun(real,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_783
tff(fact_8963_ATP_Olambda__784,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_zy(fun(A,A),fun(real,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_784
tff(fact_8964_ATP_Olambda__785,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vm(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(B,C,Uua,aa(A,set(B),Uu2,Uub)) ) ).

% ATP.lambda_785
tff(fact_8965_ATP_Olambda__786,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vl(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu2,Uub)) ) ).

% ATP.lambda_786
tff(fact_8966_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_apj(B,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu2),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_787
tff(fact_8967_ATP_Olambda__788,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apt(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_788
tff(fact_8968_ATP_Olambda__789,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apl(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_789
tff(fact_8969_ATP_Olambda__790,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ans(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_790
tff(fact_8970_ATP_Olambda__791,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ul(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_791
tff(fact_8971_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_apc(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_792
tff(fact_8972_ATP_Olambda__793,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_793
tff(fact_8973_ATP_Olambda__794,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anw(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_794
tff(fact_8974_ATP_Olambda__795,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_795
tff(fact_8975_ATP_Olambda__796,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ar(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_796
tff(fact_8976_ATP_Olambda__797,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gs(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_797
tff(fact_8977_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bz(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_798
tff(fact_8978_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ais(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_799
tff(fact_8979_ATP_Olambda__800,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_adt(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_800
tff(fact_8980_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_801
tff(fact_8981_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_802
tff(fact_8982_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zr(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_803
tff(fact_8983_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aft(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_804
tff(fact_8984_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aho(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_805
tff(fact_8985_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_afw(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_806
tff(fact_8986_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_amd(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_807
tff(fact_8987_ATP_Olambda__808,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xu(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_808
tff(fact_8988_ATP_Olambda__809,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_jd(A,fun(fun(B,nat),fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_809
tff(fact_8989_ATP_Olambda__810,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_akn(fun(A,nat),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,Uua),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_810
tff(fact_8990_ATP_Olambda__811,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wb(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_811
tff(fact_8991_ATP_Olambda__812,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ve(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_812
tff(fact_8992_ATP_Olambda__813,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wy(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_813
tff(fact_8993_ATP_Olambda__814,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xr(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_814
tff(fact_8994_ATP_Olambda__815,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uz(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_815
tff(fact_8995_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_afz(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_816
tff(fact_8996_ATP_Olambda__817,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_lt(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> dvd_dvd(B,Uua,aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_817
tff(fact_8997_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_ajb(fun(A,real),fun(nat,fun(A,real)),Uu2),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu2,Uub)) ) ).

% ATP.lambda_818
tff(fact_8998_ATP_Olambda__819,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_agd(fun(A,real),fun(nat,fun(A,real)),Uu2),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_819
tff(fact_8999_ATP_Olambda__820,axiom,
    ! [A: $tType,Uu2: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aoj($o,fun(fun(A,$o),fun(A,$o)),(Uu2)),Uua),Uub)
    <=> ( (Uu2)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_820
tff(fact_9000_ATP_Olambda__821,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_we(fun(B,set(A)),fun(B,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu2,Uua)) ) ).

% ATP.lambda_821
tff(fact_9001_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_wx(B,fun(fun(A,set(B)),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uu2),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_822
tff(fact_9002_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_vz(A,fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_823
tff(fact_9003_ATP_Olambda__824,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu2),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu2),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_824
tff(fact_9004_ATP_Olambda__825,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: 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_ys(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu2),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu2),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_825
tff(fact_9005_ATP_Olambda__826,axiom,
    ! [A: $tType,Uu2: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aoi($o,fun(fun(A,$o),fun(A,$o)),(Uu2)),Uua),Uub)
    <=> ( (Uu2)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_826
tff(fact_9006_ATP_Olambda__827,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_sl(fun(A,$o),fun(A,fun($o,$o)),Uu2),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_827
tff(fact_9007_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_sz(fun(list(A),A),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu2,Uua) ) ) ) ).

% ATP.lambda_828
tff(fact_9008_ATP_Olambda__829,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jn(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_829
tff(fact_9009_ATP_Olambda__830,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aoo(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))) ) ) ).

% ATP.lambda_830
tff(fact_9010_ATP_Olambda__831,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_831
tff(fact_9011_ATP_Olambda__832,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hb(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_832
tff(fact_9012_ATP_Olambda__833,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ajq(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_833
tff(fact_9013_ATP_Olambda__834,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aiy(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uub),Uua)) ) ).

% ATP.lambda_834
tff(fact_9014_ATP_Olambda__835,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_aeg(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_835
tff(fact_9015_ATP_Olambda__836,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zn(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_836
tff(fact_9016_ATP_Olambda__837,axiom,
    ! [Uu2: fun(real,$o),Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_apy(fun(real,$o),fun(real,fun(real,$o)),Uu2),Uua),Uub)
    <=> aa(real,$o,Uu2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ) ).

% ATP.lambda_837
tff(fact_9017_ATP_Olambda__838,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_amc(fun(real,A),fun(real,fun(real,A)),Uu2),Uua),Uub) = aa(real,A,Uu2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_838
tff(fact_9018_ATP_Olambda__839,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aqv(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_839
tff(fact_9019_ATP_Olambda__840,axiom,
    ! [A: $tType,Uu2: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_vh(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu2),Uua),Uub) = aa(nat,set(A),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_840
tff(fact_9020_ATP_Olambda__841,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_841
tff(fact_9021_ATP_Olambda__842,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aix(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_842
tff(fact_9022_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_arb(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_843
tff(fact_9023_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_844
tff(fact_9024_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hx(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_845
tff(fact_9025_ATP_Olambda__846,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_api(fun(A,$o),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_846
tff(fact_9026_ATP_Olambda__847,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_aer(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_847
tff(fact_9027_ATP_Olambda__848,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_aih(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_848
tff(fact_9028_ATP_Olambda__849,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aaj(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_849
tff(fact_9029_ATP_Olambda__850,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ok(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_850
tff(fact_9030_ATP_Olambda__851,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ou(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(product_prod(A,B),C,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_851
tff(fact_9031_ATP_Olambda__852,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_als(B,fun(fun(B,C),fun(B,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),Uub)) ) ).

% ATP.lambda_852
tff(fact_9032_ATP_Olambda__853,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aef(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),Uub)) ) ).

% ATP.lambda_853
tff(fact_9033_ATP_Olambda__854,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu2)) ) ).

% ATP.lambda_854
tff(fact_9034_ATP_Olambda__855,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_acn(fun(real,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,Uu2,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_855
tff(fact_9035_ATP_Olambda__856,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_amr(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu2),Uua),Uub) = aa(real,A,Uu2,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_856
tff(fact_9036_ATP_Olambda__857,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ara(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_857
tff(fact_9037_ATP_Olambda__858,axiom,
    ! [B: $tType,C: $tType,D6: $tType,A: $tType,Uu2: fun(D6,fun(B,C)),Uua: fun(A,D6),Uub: A] : aa(A,fun(B,C),aa(fun(A,D6),fun(A,fun(B,C)),aTP_Lamp_uc(fun(D6,fun(B,C)),fun(fun(A,D6),fun(A,fun(B,C))),Uu2),Uua),Uub) = aa(D6,fun(B,C),Uu2,aa(A,D6,Uua,Uub)) ).

% ATP.lambda_858
tff(fact_9038_ATP_Olambda__859,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,set(A)),Uua: fun(B,C),Uub: B] : aa(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_wv(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),Uu2),Uua),Uub) = aa(C,set(A),Uu2,aa(B,C,Uua,Uub)) ).

% ATP.lambda_859
tff(fact_9039_ATP_Olambda__860,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_yp(fun(C,A),fun(fun(B,C),fun(B,A)),Uu2),Uua),Uub) = aa(C,A,Uu2,aa(B,C,Uua,Uub)) ).

% ATP.lambda_860
tff(fact_9040_ATP_Olambda__861,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_sf(fun(B,A),fun(fun(num,B),fun(num,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(num,B,Uua,Uub)) ).

% ATP.lambda_861
tff(fact_9041_ATP_Olambda__862,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_rw(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_862
tff(fact_9042_ATP_Olambda__863,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_qf(fun(B,A),fun(fun(C,B),fun(C,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(C,B,Uua,Uub)) ).

% ATP.lambda_863
tff(fact_9043_ATP_Olambda__864,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu2: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_arh(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_864
tff(fact_9044_ATP_Olambda__865,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_arr(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_865
tff(fact_9045_ATP_Olambda__866,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_anu(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_866
tff(fact_9046_ATP_Olambda__867,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ajn(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_867
tff(fact_9047_ATP_Olambda__868,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alc(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_868
tff(fact_9048_ATP_Olambda__869,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acc(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_869
tff(fact_9049_ATP_Olambda__870,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_akz(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_870
tff(fact_9050_ATP_Olambda__871,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ala(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_871
tff(fact_9051_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ajo(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_872
tff(fact_9052_ATP_Olambda__873,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_873
tff(fact_9053_ATP_Olambda__874,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aam(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_874
tff(fact_9054_ATP_Olambda__875,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aen(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_875
tff(fact_9055_ATP_Olambda__876,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_aop(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,Uu2,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_876
tff(fact_9056_ATP_Olambda__877,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afa(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ).

% ATP.lambda_877
tff(fact_9057_ATP_Olambda__878,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_aeo(fun(C,A),fun(fun(A,B),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu2,Uub)) ) ).

% ATP.lambda_878
tff(fact_9058_ATP_Olambda__879,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_yq(fun(B,A),fun(fun(A,C),fun(B,C)),Uu2),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu2,Uub)) ).

% ATP.lambda_879
tff(fact_9059_ATP_Olambda__880,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_880
tff(fact_9060_ATP_Olambda__881,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aeh(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_881
tff(fact_9061_ATP_Olambda__882,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_apo(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_882
tff(fact_9062_ATP_Olambda__883,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aan(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_883
tff(fact_9063_ATP_Olambda__884,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alx(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_884
tff(fact_9064_ATP_Olambda__885,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_adw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_885
tff(fact_9065_ATP_Olambda__886,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aos(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_886
tff(fact_9066_ATP_Olambda__887,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_is(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_887
tff(fact_9067_ATP_Olambda__888,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_hw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_888
tff(fact_9068_ATP_Olambda__889,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_889
tff(fact_9069_ATP_Olambda__890,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_aip(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu2),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_aio(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_890
tff(fact_9070_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_adq(fun(nat,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_adp(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_891
tff(fact_9071_ATP_Olambda__892,axiom,
    ! [B: $tType,C: $tType,A: $tType,E4: $tType,D6: $tType,Uu2: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: set(product_prod(D6,E4)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A)),aTP_Lamp_yd(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A))),Uu2),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D6,E4)),set(set(A)),image(product_prod(D6,E4),set(A),aa(product_prod(B,C),fun(product_prod(D6,E4),set(A)),aTP_Lamp_yc(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(B,C),fun(product_prod(D6,E4),set(A))),Uu2),Uub)),Uua)) ).

% ATP.lambda_892
tff(fact_9072_ATP_Olambda__893,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_wn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_wm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub)),Uua)) ) ).

% ATP.lambda_893
tff(fact_9073_ATP_Olambda__894,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_wp(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_wm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub)),Uua)) ) ).

% ATP.lambda_894
tff(fact_9074_ATP_Olambda__895,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_amg(A,fun(set(A),fun(A,filter(A))),Uu2),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_895
tff(fact_9075_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_amf(A,fun(set(A),fun(A,filter(A))),Uu2),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_896
tff(fact_9076_ATP_Olambda__897,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A),Uub: set(A)] : aa(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_amy(A,fun(set(A),fun(set(A),filter(A))),Uu2),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_897
tff(fact_9077_ATP_Olambda__898,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_898
tff(fact_9078_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_jl(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_899
tff(fact_9079_ATP_Olambda__900,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_wl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu2,Uub)),Uua)) ) ).

% ATP.lambda_900
tff(fact_9080_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_wo(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu2,Uub)),Uua)) ) ).

% ATP.lambda_901
tff(fact_9081_ATP_Olambda__902,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aks(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu2,Uub)),Uua)) ).

% ATP.lambda_902
tff(fact_9082_ATP_Olambda__903,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apn(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_903
tff(fact_9083_ATP_Olambda__904,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aox(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_904
tff(fact_9084_ATP_Olambda__905,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_apm(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_905
tff(fact_9085_ATP_Olambda__906,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aou(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_906
tff(fact_9086_ATP_Olambda__907,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_app(A,fun(fun(C,A),fun(C,$o)),Uu2),Uua),Uub)
        <=> ( aa(C,A,Uua,Uub) != Uu2 ) ) ) ).

% ATP.lambda_907
tff(fact_9087_ATP_Olambda__908,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_wr(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),Uu2),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_908
tff(fact_9088_ATP_Olambda__909,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wq(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu2),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu2),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_909
tff(fact_9089_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_xt(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu2),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,Uu2),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_910
tff(fact_9090_ATP_Olambda__911,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_xl(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu2),Uua),Uub) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),Uu2),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_911
tff(fact_9091_ATP_Olambda__912,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_lu(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ~ dvd_dvd(B,Uua,aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_912
tff(fact_9092_ATP_Olambda__913,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_mx(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu2),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kl(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub))) ).

% ATP.lambda_913
tff(fact_9093_ATP_Olambda__914,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu2: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_adn(fun(B,real),fun(fun(real,A),fun(B,real)),Uu2),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu2,Uub)))) ) ).

% ATP.lambda_914
tff(fact_9094_ATP_Olambda__915,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A] :
      ( aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_xe(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),Uu2)
          & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),X2) ) ) ).

% ATP.lambda_915
tff(fact_9095_ATP_Olambda__916,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_rm(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [X2: A] :
          ( aa(set(A),$o,member(A,X2),Uu2)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X2) ) ) ).

% ATP.lambda_916
tff(fact_9096_ATP_Olambda__917,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_rr(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),Uua)
          & ( aa(B,option(A),Uu2,X2) = some(A,Uub) ) ) ) ).

% ATP.lambda_917
tff(fact_9097_ATP_Olambda__918,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_arg(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),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),Uu2),set_or7035219750837199246ssThan(nat,Uub,N2)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_918
tff(fact_9098_ATP_Olambda__919,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_xh(fun(B,set(A)),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),Uua)
          & aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu2,X2)) ) ) ).

% ATP.lambda_919
tff(fact_9099_ATP_Olambda__920,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ro(fun(B,A),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),Uua)
          & ( Uub = aa(B,A,Uu2,X2) ) ) ) ).

% ATP.lambda_920
tff(fact_9100_ATP_Olambda__921,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ars(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [A10: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A10)
             => ! [B11: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A10),B11)
                 => 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),Uu2),set_or3652927894154168847AtMost(nat,A10,B11)))),aa(nat,real,Uua,A10)) ) ) ) ) ).

% ATP.lambda_921
tff(fact_9101_ATP_Olambda__922,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Uu2: fun(B,fun(C,fun(D6,A))),Uua: D6,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_pb(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(D6,A,aa(C,fun(D6,A),aa(B,fun(C,fun(D6,A)),Uu2,Uub),Uuc),Uua) ).

% ATP.lambda_922
tff(fact_9102_ATP_Olambda__923,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gb(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_923
tff(fact_9103_ATP_Olambda__924,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dx(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_924
tff(fact_9104_ATP_Olambda__925,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dz(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_925
tff(fact_9105_ATP_Olambda__926,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_fm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_926
tff(fact_9106_ATP_Olambda__927,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_jp(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_927
tff(fact_9107_ATP_Olambda__928,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_ame(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu2),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_928
tff(fact_9108_ATP_Olambda__929,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_jq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_929
tff(fact_9109_ATP_Olambda__930,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_fn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_930
tff(fact_9110_ATP_Olambda__931,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_jy(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uuc),Uua),aa(A,B,Uu2,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_931
tff(fact_9111_ATP_Olambda__932,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: 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_lp(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(set(nat),$o,member(nat,Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu2,Uuc)) ) ).

% ATP.lambda_932
tff(fact_9112_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: 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_lw(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_933
tff(fact_9113_ATP_Olambda__934,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: 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_lv(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_934
tff(fact_9114_ATP_Olambda__935,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: 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_ne(A,fun(fun(A,B),fun(B,fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_935
tff(fact_9115_ATP_Olambda__936,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,$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_qa(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu2,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_936
tff(fact_9116_ATP_Olambda__937,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,$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_lr(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_937
tff(fact_9117_ATP_Olambda__938,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,$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_lq(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_938
tff(fact_9118_ATP_Olambda__939,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_adz(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu2,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_939
tff(fact_9119_ATP_Olambda__940,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_amk(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu2,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_940
tff(fact_9120_ATP_Olambda__941,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(A,$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,$o)),aTP_Lamp_nz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_941
tff(fact_9121_ATP_Olambda__942,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_rp(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu2,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_942
tff(fact_9122_ATP_Olambda__943,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hu(A,fun(A,fun(nat,fun(nat,A))),Uu2),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_ht(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_943
tff(fact_9123_ATP_Olambda__944,axiom,
    ! [Uu2: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_zi($o,fun($o,fun(code_integer,fun($o,char))),(Uu2)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_zh($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu2)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_944
tff(fact_9124_ATP_Olambda__945,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_gq(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu2),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gp(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uuc)),Uub) ) ).

% ATP.lambda_945
tff(fact_9125_ATP_Olambda__946,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fr(nat,fun(A,fun(A,fun(nat,A))),Uu2),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 = Uu2,one_one(A),zero_zero(A)) )),
            aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_946
tff(fact_9126_ATP_Olambda__947,axiom,
    ! [C: $tType,A: $tType,B: $tType,E4: $tType,D6: $tType,Uu2: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: product_prod(D6,E4),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D6,E4),fun(B,fun(C,set(A))),aTP_Lamp_yb(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(D6,E4),fun(B,fun(C,set(A)))),Uu2),Uua),Uub),Uuc) = aa(product_prod(D6,E4),set(A),aa(fun(D6,fun(E4,set(A))),fun(product_prod(D6,E4),set(A)),product_case_prod(D6,E4,set(A)),aa(C,fun(D6,fun(E4,set(A))),aa(B,fun(C,fun(D6,fun(E4,set(A)))),Uu2,Uub),Uuc)),Uua) ).

% ATP.lambda_947
tff(fact_9127_ATP_Olambda__948,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$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_ks(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_kr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_948
tff(fact_9128_ATP_Olambda__949,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$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_kp(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_kn(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_949
tff(fact_9129_ATP_Olambda__950,axiom,
    ! [Uu2: 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_fi(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),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_fh(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_950
tff(fact_9130_ATP_Olambda__951,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_fd(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu2),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_fc(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_951
tff(fact_9131_ATP_Olambda__952,axiom,
    ! [Uu2: 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_abc(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,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_952
tff(fact_9132_ATP_Olambda__953,axiom,
    ! [Uu2: 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_aba(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,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_953
tff(fact_9133_ATP_Olambda__954,axiom,
    ! [Uu2: 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_aaz(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_954
tff(fact_9134_ATP_Olambda__955,axiom,
    ! [Uu2: 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_aay(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_955
tff(fact_9135_ATP_Olambda__956,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hf(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(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),Uu2),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu2),Uuc))),aa(nat,A,power_power(A,Uu2),Uub)) ) ).

% ATP.lambda_956
tff(fact_9136_ATP_Olambda__957,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ht(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),Uu2)),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_957
tff(fact_9137_ATP_Olambda__958,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: 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_akp(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu2,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_958
tff(fact_9138_ATP_Olambda__959,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_yv(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_959
tff(fact_9139_ATP_Olambda__960,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu2: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_aon(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,aa(A,B,Uu2,Uuc)),Uua)
            & ( aa(A,B,Uu2,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_960
tff(fact_9140_ATP_Olambda__961,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ape(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,aa(A,B,Uu2,Uuc)),Uub)
            & ( aa(A,B,Uu2,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_961
tff(fact_9141_ATP_Olambda__962,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqt(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_962
tff(fact_9142_ATP_Olambda__963,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aqf(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_963
tff(fact_9143_ATP_Olambda__964,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_964
tff(fact_9144_ATP_Olambda__965,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ew(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),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,Uu2),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_965
tff(fact_9145_ATP_Olambda__966,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ff(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_966
tff(fact_9146_ATP_Olambda__967,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fq(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uub),Uuc)))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_967
tff(fact_9147_ATP_Olambda__968,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hi(A,fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu2),Uuc)) ) ).

% ATP.lambda_968
tff(fact_9148_ATP_Olambda__969,axiom,
    ! [A: $tType,Uu2: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_cz($o,fun(A,fun(A,fun(A,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( ( (Uu2)
         => ( Uuc = Uua ) )
        & ( ~ (Uu2)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_969
tff(fact_9149_ATP_Olambda__970,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_qg(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),aa(set(B),set(A),image(B,A,Uu2),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_970
tff(fact_9150_ATP_Olambda__971,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: fun(A,fun(B,$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_kl(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_971
tff(fact_9151_ATP_Olambda__972,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,fun(B,$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_ko(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_972
tff(fact_9152_ATP_Olambda__973,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hg(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu2),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu2),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_973
tff(fact_9153_ATP_Olambda__974,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_aol(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_974
tff(fact_9154_ATP_Olambda__975,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_qe(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_975
tff(fact_9155_ATP_Olambda__976,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_zc(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu2)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_976
tff(fact_9156_ATP_Olambda__977,axiom,
    ! [A: $tType,C: $tType,Uu2: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_qj(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_977
tff(fact_9157_ATP_Olambda__978,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_qn(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_978
tff(fact_9158_ATP_Olambda__979,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hh(A,fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu2),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_979
tff(fact_9159_ATP_Olambda__980,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_eu(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu2),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_980
tff(fact_9160_ATP_Olambda__981,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pi(int,fun(int,fun(int,fun(int,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_981
tff(fact_9161_ATP_Olambda__982,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pf(int,fun(int,fun(int,fun(int,$o))),Uu2),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),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_982
tff(fact_9162_ATP_Olambda__983,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_oj(A,fun(B,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_983
tff(fact_9163_ATP_Olambda__984,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_qh(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & aa(A,$o,Uub,aa(B,A,Uu2,Uuc)) ) ) ).

% ATP.lambda_984
tff(fact_9164_ATP_Olambda__985,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_wh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu2,Uua)) ) ).

% ATP.lambda_985
tff(fact_9165_ATP_Olambda__986,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_kw(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
            & ( 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_986
tff(fact_9166_ATP_Olambda__987,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ku(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu2)
            & ( 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_987
tff(fact_9167_ATP_Olambda__988,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_akx(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_988
tff(fact_9168_ATP_Olambda__989,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: 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_bh(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_989
tff(fact_9169_ATP_Olambda__990,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: 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_nq(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_990
tff(fact_9170_ATP_Olambda__991,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: 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_adp(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,power_power(A,Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_991
tff(fact_9171_ATP_Olambda__992,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu2: 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_aio(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),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,Uu2,Uub)),Uuc)) ) ).

% ATP.lambda_992
tff(fact_9172_ATP_Olambda__993,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_ade(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu2,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_993
tff(fact_9173_ATP_Olambda__994,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_ik(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_994
tff(fact_9174_ATP_Olambda__995,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aqm(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(set(B),$o,member(B,aa(A,B,Uu2,Uuc)),aa(set(B),set(B),minus_minus(set(B),Uub),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_995
tff(fact_9175_ATP_Olambda__996,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: 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_po(fun(B,A),fun(A,fun(B,fun(A,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_996
tff(fact_9176_ATP_Olambda__997,axiom,
    ! [Uu2: 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_fh(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_997
tff(fact_9177_ATP_Olambda__998,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: 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_ey(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_998
tff(fact_9178_ATP_Olambda__999,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_fc(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_999
tff(fact_9179_ATP_Olambda__1000,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gp(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1000
tff(fact_9180_ATP_Olambda__1001,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_xj(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu2),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu2,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1001
tff(fact_9181_ATP_Olambda__1002,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_vc(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu2,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1002
tff(fact_9182_ATP_Olambda__1003,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_xa(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu2),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu2,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1003
tff(fact_9183_ATP_Olambda__1004,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ux(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu2,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1004
tff(fact_9184_ATP_Olambda__1005,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_ta(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu2,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_1005
tff(fact_9185_ATP_Olambda__1006,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$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_kq(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uuc),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kl(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_1006
tff(fact_9186_ATP_Olambda__1007,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$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_km(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kl(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_1007
tff(fact_9187_ATP_Olambda__1008,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_adi(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,cos(real,aa(A,real,Uu2,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_1008
tff(fact_9188_ATP_Olambda__1009,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_adg(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu2,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_1009
tff(fact_9189_ATP_Olambda__1010,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_acf(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu2,Uub))) ) ).

% ATP.lambda_1010
tff(fact_9190_ATP_Olambda__1011,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ach(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu2,Uub))) ) ).

% ATP.lambda_1011
tff(fact_9191_ATP_Olambda__1012,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_acu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),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,Uu2,Uua))) ) ).

% ATP.lambda_1012
tff(fact_9192_ATP_Olambda__1013,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_abj(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_1013
tff(fact_9193_ATP_Olambda__1014,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_acq(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu2,Uub)))) ) ).

% ATP.lambda_1014
tff(fact_9194_ATP_Olambda__1015,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_adm(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_1015
tff(fact_9195_ATP_Olambda__1016,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_aky(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu2,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_1016
tff(fact_9196_ATP_Olambda__1017,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_alf(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub))) ) ).

% ATP.lambda_1017
tff(fact_9197_ATP_Olambda__1018,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_aob(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( aa(set(A),$o,member(A,Uuc),Uub)
             => aa(A,$o,Uu2,Uuc) ) ) ) ) ).

% ATP.lambda_1018
tff(fact_9198_ATP_Olambda__1019,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_aql(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$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,Uu2,Uuc))),Uub)) ) ) ).

% ATP.lambda_1019
tff(fact_9199_ATP_Olambda__1020,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_alj(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_abu(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_abu(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_abu(A,A)))))) ) ).

% ATP.lambda_1020
tff(fact_9200_ATP_Olambda__1021,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_alh(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu2,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_1021
tff(fact_9201_ATP_Olambda__1022,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_alg(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_1022
tff(fact_9202_ATP_Olambda__1023,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_qx(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_qn(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_1023
tff(fact_9203_ATP_Olambda__1024,axiom,
    ! [A: $tType,Uu2: 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_si(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu2),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu2),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_1024
tff(fact_9204_ATP_Olambda__1025,axiom,
    ! [A: $tType,B: $tType,Uu2: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ox($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1025
tff(fact_9205_ATP_Olambda__1026,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_qm(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_qj(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_1026
tff(fact_9206_ATP_Olambda__1027,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_qp(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_qn(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_1027
tff(fact_9207_ATP_Olambda__1028,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_qk(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_qj(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_1028
tff(fact_9208_ATP_Olambda__1029,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_qo(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_qn(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_1029
tff(fact_9209_ATP_Olambda__1030,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_aqd(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$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_bi(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua)))) ) ) ).

% ATP.lambda_1030
tff(fact_9210_ATP_Olambda__1031,axiom,
    ! [A: $tType,C: $tType,B: $tType,D6: $tType,Uu2: fun(B,A),Uua: fun(C,fun(D6,B)),Uub: C,Uuc: D6] : aa(D6,A,aa(C,fun(D6,A),aa(fun(C,fun(D6,B)),fun(C,fun(D6,A)),aTP_Lamp_os(fun(B,A),fun(fun(C,fun(D6,B)),fun(C,fun(D6,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,Uu2,aa(D6,B,aa(C,fun(D6,B),Uua,Uub),Uuc)) ).

% ATP.lambda_1031
tff(fact_9211_ATP_Olambda__1032,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_lj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(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_1032
tff(fact_9212_ATP_Olambda__1033,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_lh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(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_1033
tff(fact_9213_ATP_Olambda__1034,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_jg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1034
tff(fact_9214_ATP_Olambda__1035,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_hy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1035
tff(fact_9215_ATP_Olambda__1036,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_xb(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu2),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_xa(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu2),Uua),Uuc)),Uub)) ).

% ATP.lambda_1036
tff(fact_9216_ATP_Olambda__1037,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_uy(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu2),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ux(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1037
tff(fact_9217_ATP_Olambda__1038,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_xk(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu2),Uua),Uub),Uuc) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_xj(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu2),Uua),Uuc)),Uub)) ).

% ATP.lambda_1038
tff(fact_9218_ATP_Olambda__1039,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_vd(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu2),Uua),Uub),Uuc) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_vc(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1039
tff(fact_9219_ATP_Olambda__1040,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1040
tff(fact_9220_ATP_Olambda__1041,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1041
tff(fact_9221_ATP_Olambda__1042,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: 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_acm(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu2),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,Uu2,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)))) ) ).

% ATP.lambda_1042
tff(fact_9222_ATP_Olambda__1043,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1043
tff(fact_9223_ATP_Olambda__1044,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_1044
tff(fact_9224_ATP_Olambda__1045,axiom,
    ! [C: $tType,A: $tType,B: $tType,E4: $tType,D6: $tType,Uu2: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: set(product_prod(D6,E4)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D6,E4)),fun(B,fun(C,set(A))),aTP_Lamp_ya(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(B,fun(C,set(A)))),Uu2),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D6,E4)),set(set(A)),image(product_prod(D6,E4),set(A),aa(fun(D6,fun(E4,set(A))),fun(product_prod(D6,E4),set(A)),product_case_prod(D6,E4,set(A)),aa(C,fun(D6,fun(E4,set(A))),aa(B,fun(C,fun(D6,fun(E4,set(A)))),Uu2,Uub),Uuc))),Uua)) ).

% ATP.lambda_1045
tff(fact_9225_ATP_Olambda__1046,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_arn(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu2),Uua),Uub),Uuc)
    <=> ! [X2: A] :
          ( ( aa(set(A),$o,member(A,X2),Uu2)
           => aa(set(B),$o,member(B,aa(A,B,Uuc,X2)),Uua) )
          & ( ~ aa(set(A),$o,member(A,X2),Uu2)
           => ( aa(A,B,Uuc,X2) = Uub ) ) ) ) ).

% ATP.lambda_1046
tff(fact_9226_ATP_Olambda__1047,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_acd(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu2,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_1047
tff(fact_9227_ATP_Olambda__1048,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_ada(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu2),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_acz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud)),Uu2) ) ).

% ATP.lambda_1048
tff(fact_9228_ATP_Olambda__1049,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hr(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),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_hq(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_1049
tff(fact_9229_ATP_Olambda__1050,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_zh($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu2)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_zg($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu2)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1050
tff(fact_9230_ATP_Olambda__1051,axiom,
    ! [Uu2: 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_abb(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu2),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_aba(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu2),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uud),aa(nat,nat,minus_minus(nat,Uu2),Uuc))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu2),Uuc)))))) ).

% ATP.lambda_1051
tff(fact_9231_ATP_Olambda__1052,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_il(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),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_ik(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu2))),aa(nat,A,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_1052
tff(fact_9232_ATP_Olambda__1053,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_fu(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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,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,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_1053
tff(fact_9233_ATP_Olambda__1054,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_fo(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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),Uu2)),Uua)),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_1054
tff(fact_9234_ATP_Olambda__1055,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_fp(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_1055
tff(fact_9235_ATP_Olambda__1056,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hq(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu2),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,Uu2,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_1056
tff(fact_9236_ATP_Olambda__1057,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_apv(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uud),Uua)) ) ) ).

% ATP.lambda_1057
tff(fact_9237_ATP_Olambda__1058,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_aqe(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> ( ( aa(A,B,Uu2,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_1058
tff(fact_9238_ATP_Olambda__1059,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_acs(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu2),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_1059
tff(fact_9239_ATP_Olambda__1060,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_ra(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),Uuc) ),
        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub),
        $ite(
          ( ( Uuc = Uud )
          & ( Uuc = Uue ) ),
          vEBT_Node(none(product_prod(nat,nat)),Uu2,Uua,Uub),
          $let(
            xn: nat,
            xn:= 
              $ite(Uuc = Uud,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Uub))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Uub)))))),Uuc),
            $let(
              minn: nat,
              minn:= 
                $ite(Uuc = Uud,xn,Uud),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),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)),Uua)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,Uua,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Uub,h),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Uue,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                  Uue ))),Uu2,newlist,sn) ),
                        vEBT_Node(some(product_prod(nat,nat),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Uue,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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Uue))),Uu2,newlist,Uub) ) ) ),
                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub) ) ) ) ) ) ) ).

% ATP.lambda_1060
tff(fact_9240_ATP_Olambda__1061,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,option(nat),aa(nat,fun(nat,option(nat)),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_rg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),
        some(nat,Uud),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),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)),Uua)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),some(nat,h)),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Uub,h),
                    $ite(sc = none(nat),none(nat),vEBT_VEBT_add(vEBT_VEBT_mul(some(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),sc),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% ATP.lambda_1061
tff(fact_9241_ATP_Olambda__1062,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Uua: nat,Uub: list(vEBT_VEBT),Uuc: vEBT_VEBT,Uud: nat,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),vEBT_VEBT,aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_rb(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),one_one(nat)),vEBT_Node(Uu2,Uua,Uub,Uuc),aa(product_prod(nat,nat),vEBT_VEBT,aa(fun(nat,fun(nat,vEBT_VEBT)),fun(product_prod(nat,nat),vEBT_VEBT),product_case_prod(nat,nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_ra(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uua),Uub),Uuc),Uud)),Uue)) ).

% ATP.lambda_1062
tff(fact_9242_ATP_Olambda__1063,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_zg($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu2)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_zf($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1063
tff(fact_9243_ATP_Olambda__1064,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_rd(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uud,Uuc),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),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)),Uua))
            & ~ ( ( Uuc = Uud )
                | ( Uuc = Uue ) ) ),
            vEBT_Node(some(product_prod(nat,nat),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uuc,Uud)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Uue))),Uu2,list_update(vEBT_VEBT,Uua,h,aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),aa(nat,vEBT_VEBT,aa(vEBT_VEBT,fun(nat,vEBT_VEBT),vEBT_vebt_insert,Uub),h),Uub)),
            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub) ) ) ) ).

% ATP.lambda_1064
tff(fact_9244_ATP_Olambda__1065,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_acz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),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)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_acx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),minus_minus(set(A),Uu2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_1065
tff(fact_9245_ATP_Olambda__1066,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: 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_acj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,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_1066
tff(fact_9246_ATP_Olambda__1067,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_adc(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu2,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,Uu2,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu2,Uub)))) ) ).

% ATP.lambda_1067
tff(fact_9247_ATP_Olambda__1068,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_abo(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),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,Uu2,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1068
tff(fact_9248_ATP_Olambda__1069,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: 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_abm(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),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,Uu2,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_1069
tff(fact_9249_ATP_Olambda__1070,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: 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_acw(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),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,Uu2,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))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1070
tff(fact_9250_ATP_Olambda__1071,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_zf($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_ze($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1071
tff(fact_9251_ATP_Olambda__1072,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_zd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu2),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1072
tff(fact_9252_ATP_Olambda__1073,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_ze($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_zd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1073
tff(fact_9253_ATP_Olambda__1074,axiom,
    ! [B: $tType,A: $tType,Uu2: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_yx($o,fun(A,fun(B,$o)),(Uu2)),Uua),Uub)
    <=> (Uu2) ) ).

% ATP.lambda_1074
tff(fact_9254_ATP_Olambda__1075,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: $o,Uua: A] :
          ( aa(A,$o,aTP_Lamp_alm($o,fun(A,$o),(Uu2)),Uua)
        <=> (Uu2) ) ) ).

% ATP.lambda_1075
tff(fact_9255_ATP_Olambda__1076,axiom,
    ! [A: $tType,Uu2: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_yf($o,fun(A,$o),(Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_1076
tff(fact_9256_ATP_Olambda__1077,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_vy(set(A),fun(B,set(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_1077
tff(fact_9257_ATP_Olambda__1078,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_uv(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1078
tff(fact_9258_ATP_Olambda__1079,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_xx(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1079
tff(fact_9259_ATP_Olambda__1080,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_abt(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1080
tff(fact_9260_ATP_Olambda__1081,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_agr(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1081
tff(fact_9261_ATP_Olambda__1082,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_adr(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1082
tff(fact_9262_ATP_Olambda__1083,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_vx(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1083
tff(fact_9263_ATP_Olambda__1084,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_aqz(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1084
tff(fact_9264_ATP_Olambda__1085,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_aek(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1085
tff(fact_9265_ATP_Olambda__1086,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_afb(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1086
tff(fact_9266_ATP_Olambda__1087,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_qc(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1087
tff(fact_9267_ATP_Olambda__1088,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_qi(B,fun(A,B),Uu2),Uua) = Uu2 ).

% ATP.lambda_1088
tff(fact_9268_ATP_Olambda__1089,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_xz(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1089
tff(fact_9269_ATP_Olambda__1090,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ax(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1090
tff(fact_9270_ATP_Olambda__1091,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_aaa(A,fun(A,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1091
tff(fact_9271_ATP_Olambda__1092,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_xp(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1092
tff(fact_9272_ATP_Olambda__1093,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_nb(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1093
tff(fact_9273_ATP_Olambda__1094,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_aje(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1094
tff(fact_9274_ATP_Olambda__1095,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_aem(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1095
tff(fact_9275_ATP_Olambda__1096,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_mr(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1096
tff(fact_9276_ATP_Olambda__1097,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_mt(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1097
tff(fact_9277_ATP_Olambda__1098,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ari(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1098
tff(fact_9278_ATP_Olambda__1099,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_aei(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_1099
tff(fact_9279_ATP_Olambda__1100,axiom,
    ! [A: $tType,Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_pv(A,fun(nat,A),Uu2),Uua) = Uu2 ).

% ATP.lambda_1100
tff(fact_9280_ATP_Olambda__1101,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_pu(A,fun(B,A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_1101
tff(fact_9281_ATP_Olambda__1102,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_ud(B,fun(A,A)),Uu2),Uua) = Uua ).

% ATP.lambda_1102
tff(fact_9282_ATP_Olambda__1103,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_oy(A,fun(B,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_1103
tff(fact_9283_ATP_Olambda__1104,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ob(A,fun(A,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_1104
tff(fact_9284_ATP_Olambda__1105,axiom,
    ! [Uu2: complex] : aa(complex,complex,aTP_Lamp_gj(complex,complex),Uu2) = Uu2 ).

% ATP.lambda_1105
tff(fact_9285_ATP_Olambda__1106,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_ig(nat,nat),Uu2) = Uu2 ).

% ATP.lambda_1106
tff(fact_9286_ATP_Olambda__1107,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_in(int,int),Uu2) = Uu2 ).

% ATP.lambda_1107
tff(fact_9287_ATP_Olambda__1108,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_abu(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1108
tff(fact_9288_ATP_Olambda__1109,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ads(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1109
tff(fact_9289_ATP_Olambda__1110,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_zv(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1110
tff(fact_9290_ATP_Olambda__1111,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_adx(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1111
tff(fact_9291_ATP_Olambda__1112,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_alp(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1112
tff(fact_9292_ATP_Olambda__1113,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_nu(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1113
tff(fact_9293_ATP_Olambda__1114,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_vt(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1114
tff(fact_9294_ATP_Olambda__1115,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_vv(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1115
tff(fact_9295_ATP_Olambda__1116,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_yr(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1116
tff(fact_9296_ATP_Olambda__1117,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_an(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_1117
tff(fact_9297_ATP_Olambda__1118,axiom,
    ! [A: $tType,Uu2: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu2) = Uu2 ).

% ATP.lambda_1118
tff(fact_9298_ATP_Olambda__1119,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_yh(B,A),Uu2) = top_top(A) ) ).

% ATP.lambda_1119
tff(fact_9299_ATP_Olambda__1120,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_yg(B,A),Uu2) = top_top(A) ) ).

% ATP.lambda_1120
tff(fact_9300_ATP_Olambda__1121,axiom,
    ! [C: $tType,B: $tType,Uu2: C] : aa(C,set(B),aTP_Lamp_uq(C,set(B)),Uu2) = bot_bot(set(B)) ).

% ATP.lambda_1121
tff(fact_9301_ATP_Olambda__1122,axiom,
    ! [B: $tType,A: $tType,Uu2: B] : aa(B,set(A),aTP_Lamp_ww(B,set(A)),Uu2) = bot_bot(set(A)) ).

% ATP.lambda_1122
tff(fact_9302_ATP_Olambda__1123,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_xy(B,A),Uu2) = bot_bot(A) ) ).

% ATP.lambda_1123
tff(fact_9303_ATP_Olambda__1124,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_vw(B,A),Uu2) = bot_bot(A) ) ).

% ATP.lambda_1124
tff(fact_9304_ATP_Olambda__1125,axiom,
    ! [A: $tType,D6: $tType,Uu2: A] : aa(A,set(D6),aTP_Lamp_ur(A,set(D6)),Uu2) = bot_bot(set(D6)) ).

% ATP.lambda_1125
tff(fact_9305_ATP_Olambda__1126,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_av(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_1126
tff(fact_9306_ATP_Olambda__1127,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_bj(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_1127
tff(fact_9307_ATP_Olambda__1128,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_gf(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_1128
tff(fact_9308_ATP_Olambda__1129,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_nt(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_1129
tff(fact_9309_ATP_Olambda__1130,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_abp(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_1130
tff(fact_9310_ATP_Olambda__1131,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_am(A,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_1131
tff(fact_9311_ATP_Olambda__1132,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_pc(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_1132
tff(fact_9312_ATP_Olambda__1133,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_io(B,A),Uu2) = one_one(A) ) ).

% ATP.lambda_1133
tff(fact_9313_ATP_Olambda__1134,axiom,
    ! [A: $tType,Uu2: A] : aa(A,real,aTP_Lamp_nd(A,real),Uu2) = one_one(real) ).

% ATP.lambda_1134
tff(fact_9314_ATP_Olambda__1135,axiom,
    ! [A: $tType,Uu2: A] : aa(A,nat,aTP_Lamp_my(A,nat),Uu2) = one_one(nat) ).

% ATP.lambda_1135
tff(fact_9315_ATP_Olambda__1136,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_cv(real,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_1136
tff(fact_9316_ATP_Olambda__1137,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_sd(nat,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_1137
tff(fact_9317_ATP_Olambda__1138,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_dc(A,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_1138
tff(fact_9318_ATP_Olambda__1139,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_se(nat,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_1139
tff(fact_9319_ATP_Olambda__1140,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_rc(A,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_1140
tff(fact_9320_ATP_Olambda__1141,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(nat,nat),aTP_Lamp_sm(A,fun(nat,nat)),Uu2) = suc ).

% ATP.lambda_1141

% Type constructors (779)
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => condit1219197933456340205attice(fun(A15,A16)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( counta3822494911875563373attice(A16)
     => counta3822494911875563373attice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple592849572758109894attice(A16)
     => comple592849572758109894attice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple489889107523837845lgebra(A16)
     => comple489889107523837845lgebra(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple6319245703460814977attice(fun(A15,A16)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( boolea8198339166811842893lgebra(A16)
     => boolea8198339166811842893lgebra(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple9053668089753744459l_ccpo(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_sup(A16)
     => semilattice_sup(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_inf(A16)
     => semilattice_inf(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A15: $tType,A16: $tType] :
      ( complete_Sup(A16)
     => complete_Sup(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A15: $tType,A16: $tType] :
      ( complete_Inf(A16)
     => complete_Inf(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_top(A16)
     => order_top(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_bot(A16)
     => order_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( preorder(A16)
     => preorder(fun(A15,A16)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( lattice(A16)
     => lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order(A16)
     => order(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A15: $tType,A16: $tType] :
      ( top(A16)
     => top(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ord(A16)
     => ord(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bot(A16)
     => bot(fun(A15,A16)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( uminus(A16)
     => uminus(fun(A15,A16)) ) ).

tff(tcon_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___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(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___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___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___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___Complete__Lattices_OSup_4,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_5,axiom,
    complete_Inf(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_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___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_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__mult_32,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_42,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_43,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_44,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_45,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_46,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_47,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_48,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_49,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_50,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_51,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_52,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_53,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_54,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_55,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_56,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_57,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_58,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_59,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_60,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_61,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_62,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_63,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_64,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_65,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_66,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_67,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_68,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_69,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_70,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_71,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_72,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_73,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_74,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_75,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_76,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_77,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_78,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_79,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_80,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_81,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_82,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_83,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_84,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_85,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_86,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_87,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_88,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_89,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_90,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_91,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_92,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_93,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_94,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_95,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_96,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_97,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Power_Opower_98,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_99,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_100,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_101,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_102,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_103,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_104,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_105,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_106,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_107,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_108,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_109,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_110,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_111,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_112,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_113,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_114,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_115,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_116,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_117,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_118,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_119,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_120,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_121,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_122,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_123,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_124,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_125,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_126,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_127,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_128,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_129,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_130,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_131,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_132,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_133,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_134,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_135,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_136,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_137,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_138,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_139,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_140,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_141,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_142,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_143,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_144,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_145,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_146,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_147,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_148,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_149,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_150,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_151,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_152,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_153,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_154,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_155,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_156,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_157,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_158,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_159,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_160,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_161,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_162,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_163,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_164,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_165,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_166,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_167,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_168,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_169,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_170,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_171,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_172,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_173,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_174,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_175,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_176,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_177,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_178,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_179,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_180,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_181,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_182,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_183,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_184,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_185,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_186,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_187,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_188,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_189,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_190,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_191,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_192,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_193,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_194,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_195,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_196,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(set(A15)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_197,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_198,axiom,
    ! [A15: $tType] : comple592849572758109894attice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_199,axiom,
    ! [A15: $tType] : comple489889107523837845lgebra(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_200,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(set(A15)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_201,axiom,
    ! [A15: $tType] : boolea8198339166811842893lgebra(set(A15)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_202,axiom,
    ! [A15: $tType] : comple9053668089753744459l_ccpo(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_203,axiom,
    ! [A15: $tType] : semilattice_sup(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_204,axiom,
    ! [A15: $tType] : semilattice_inf(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_205,axiom,
    ! [A15: $tType] : complete_Sup(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_206,axiom,
    ! [A15: $tType] : complete_Inf(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
    ! [A15: $tType] : order_top(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
    ! [A15: $tType] : order_bot(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
    ! [A15: $tType] : preorder(set(A15)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_210,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(set(A15)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_211,axiom,
    ! [A15: $tType] : lattice(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder_212,axiom,
    ! [A15: $tType] : order(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Otop_213,axiom,
    ! [A15: $tType] : top(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oord_214,axiom,
    ! [A15: $tType] : ord(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Obot_215,axiom,
    ! [A15: $tType] : bot(set(A15)) ).

tff(tcon_Set_Oset___Groups_Ouminus_216,axiom,
    ! [A15: $tType] : uminus(set(A15)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_217,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_218,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_219,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_220,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_221,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_222,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_223,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_227,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_228,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_229,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_230,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_231,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_232,axiom,
    complete_Sup($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_233,axiom,
    complete_Inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_234,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_235,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_236,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_237,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_238,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_239,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_240,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_241,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_242,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_243,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_244,axiom,
    uminus($o) ).

tff(tcon_List_Olist___Nat_Osize_245,axiom,
    ! [A15: $tType] : size(list(A15)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_246,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_247,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_248,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_249,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_250,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_251,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_252,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_253,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_254,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_255,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_256,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_257,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_258,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_259,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_260,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_261,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_262,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_263,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_264,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_265,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_266,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_267,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_268,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_269,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_270,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_271,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_272,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_273,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_274,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_275,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_276,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_277,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_278,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_279,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_280,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_281,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_282,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_283,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_284,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_285,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_286,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_287,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_288,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_289,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_290,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_291,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_292,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_293,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_294,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_295,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_296,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_297,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_298,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_299,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_300,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_301,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_302,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_303,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_304,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_305,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_306,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OSup_307,axiom,
    complete_Sup(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OInf_308,axiom,
    complete_Inf(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_312,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_313,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_314,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_315,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_316,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_317,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_318,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_319,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_320,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_321,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_322,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_323,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_324,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_325,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_326,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_327,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_328,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_329,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_330,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_331,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_332,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_333,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_334,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_335,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_336,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_337,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_338,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_339,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_340,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_341,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_342,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_343,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_344,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_345,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_346,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_347,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_348,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_349,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_350,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_351,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_352,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_353,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_354,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Nat_Osize_355,axiom,
    size(char) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_356,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_357,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_358,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_359,axiom,
    ! [A15: $tType] : comple9053668089753744459l_ccpo(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_360,axiom,
    ! [A15: $tType] : semilattice_sup(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_361,axiom,
    ! [A15: $tType] : semilattice_inf(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_362,axiom,
    ! [A15: $tType] : complete_Sup(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_363,axiom,
    ! [A15: $tType] : complete_Inf(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_364,axiom,
    ! [A15: $tType] : order_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_365,axiom,
    ! [A15: $tType] : order_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_366,axiom,
    ! [A15: $tType] : preorder(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_367,axiom,
    ! [A15: $tType] : lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_368,axiom,
    ! [A15: $tType] : order(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_369,axiom,
    ! [A15: $tType] : top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_370,axiom,
    ! [A15: $tType] : ord(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_371,axiom,
    ! [A15: $tType] : bot(filter(A15)) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_372,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(option(A15)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_373,axiom,
    ! [A15: $tType] : size(option(A15)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_374,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_375,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_376,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_377,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_378,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_379,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_380,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_381,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_382,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_383,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_384,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_385,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_386,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_387,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_388,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_389,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_390,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_391,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_392,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_393,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_394,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_395,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_396,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_397,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_398,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_399,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_400,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_401,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_402,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_403,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_404,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_405,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_406,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_407,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_408,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_409,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_410,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_411,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_412,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_413,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_414,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_415,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_416,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_417,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_418,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_419,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_420,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_421,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_422,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_423,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_424,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_425,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_426,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_427,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_428,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_429,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_430,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_431,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_432,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_433,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_434,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_435,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_436,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_437,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_438,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_439,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_440,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_441,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_442,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_443,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_444,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_445,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_446,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_447,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_448,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_449,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_450,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_451,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_452,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_453,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_454,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_455,axiom,
    canoni5634975068530333245id_add(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_456,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_457,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_458,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_459,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_460,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_461,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_462,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_463,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_464,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_465,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_466,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_467,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_468,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_469,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_470,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_471,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_472,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_473,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_474,axiom,
    complete_Sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_475,axiom,
    complete_Inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_476,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_477,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_478,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_479,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_480,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_481,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_482,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_483,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_484,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_485,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_486,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_487,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_488,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_489,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_490,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_491,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_492,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_493,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_494,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_495,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_496,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_497,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_498,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_499,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_500,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_501,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_502,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_503,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_504,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topolo4958980785337419405_space(A15)
        & topolo4958980785337419405_space(A16) )
     => topolo4958980785337419405_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_505,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t2_space(A15)
        & topological_t2_space(A16) )
     => topological_t2_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_506,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t1_space(A15)
        & topological_t1_space(A16) )
     => topological_t1_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_507,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_508,axiom,
    ! [A15: $tType,A16: $tType] : size(product_prod(A15,A16)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_509,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_510,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_511,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_512,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_513,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_514,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_515,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_516,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_517,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_518,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_519,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_520,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_521,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_522,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_523,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_524,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_525,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_526,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_527,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_528,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_529,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_530,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_531,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_532,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_533,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_534,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_535,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_536,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_537,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_538,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_539,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_540,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_541,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_542,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_543,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_544,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_545,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_546,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_547,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_548,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_549,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_550,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_551,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_552,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_553,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_554,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_555,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_556,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_557,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_558,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_559,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_560,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_561,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_562,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_563,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_564,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_565,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_566,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_567,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_568,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_569,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_570,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_571,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_572,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_573,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_574,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_575,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_576,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_577,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_578,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_579,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_580,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_581,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_582,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_583,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_584,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_585,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_586,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_587,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_588,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_589,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_590,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_591,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_592,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_593,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_594,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_595,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_596,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_597,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_598,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_599,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_600,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_601,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_602,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_603,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_604,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_605,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_606,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_607,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_608,axiom,
    size(vEBT_VEBT) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X9: A] : aa(A,$o,P,X9) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (3)
tff(conj_0,hypothesis,
    ! [X3: nat] :
      ( aa(set(nat),$o,member(nat,X3),aa(list(nat),set(nat),set2(nat),xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n)) ) ).

tff(conj_1,hypothesis,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),n) ).

tff(conj_2,conjecture,
    vEBT_Intf_test(n,xs,ys) = aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_aa(nat,nat)),ys) ).

%------------------------------------------------------------------------------